Lame's wave functions of the ellipsoid of revolution

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Material Information

Title:
Lame's wave functions of the ellipsoid of revolution
Series Title:
TM
Physical Description:
102 p. : ; 27 cm.
Language:
English
Creator:
Meixner, J
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Lamé's functions   ( lcsh )
Genre:
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographic references (p. 83-84).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by J. Meixner.
General Note:
"Report No. NACA TM 1224."
General Note:
"Report date April 1949."
General Note:
"Translation of "Die Laméschen Wellenfunktionen des Drehellipsoids." Zentrale für wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht Nr. 1952, June 1944."

Record Information

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003760211
oclc - 85851306
sobekcm - AA00006232_00001
System ID:
AA00006232:00001

Full Text
OAT 12-zZ










;.AAi TIM :t. 1.

T aBLE OF C 0 NT E N TS


Pope

1. INTRODUCTI:O ........... ... 1

2. TIE BASIC EQU-.TION . . .. 3
2.1 F:o-atlonailly Symmetrical Elliptic Ccrd: :* ~- 3
2.2 Sep..ration of the Inve Equation .... .... 4
2.'3 R tuction to a Differential Equation .
2.4 Trirrsformations of the Basic Eqi'ation .
2.5 Co:ur.etion with Mathie's Fur.tions . 6

. SPHERICAL A i CYLINDPIC.L FUJCTIOES .. 7
3.1 Fcw Formulas for Spherical Functions 7
3.2 A iLW Estimates for SphesicL l I'unctions .. 8
3.3 A Few Formulas fo:r Cylinari 'cal Functiorns .. .1
3.4 An 'Etimate for" Cylindrical Functions 11

4. TIHE X- FUICT.IO OF TEE FLRET i.D SECO(iT KTlD 13
4.1 DEf.:.ition of thle X-Functions uf the' Fi:.-t 'nil
Second Ki.nd ................. 1?
4.2 General Qualit-ec of the Coefficiernt: a, ... 15
4.3 Convergence of the juries LevelD;mvlnt2 of the
X-Functions of the Pirst and SeconL Kin" ... ... 16
'4.4 Further Solu.tio-;s of the Basic LEqLation anrl! Their
Relation to the X-r'uncrtiuns of the F'.rs. and
Second Kind . .. 17
4.5 General Relations between the j-Fu'nrr..f;ins .. 2.

5. TEE Z-FUfICTIONS OF TPE FIRST TO FOLU-.TH KIID . 21
5.1 DefLa'tion of the Z-Fun;ctiors of t.he First to
Four'zh Kir.. ............. ... 21
5P2 Con-er-nenco of the Series Dev.eloriment of the
Z-7unijcticns of the First and i'econd Kin .. 22
5.3 General Relations bet-.reCr: the Z-Fiuctions ..... 23
5.4 -An,'nmp,;otic Dj)celoper~sts of the Z-Functions 24
55 Fuirt!.er Solutions cf the Basic Eq2otion an. Their
Relation to the Z-Functions of tho FIrest nd
Second Kind . ... ...... 27
5.6 Lau:-rent-Develo]ments fcr 2-- e.nd Z-Functione 23
57 Corinecbion between the Z- and Z-Functions .... 31
5.8 Wron.ki'e Determin nt .............. 33
5-9 Other Series Developments of the Solutions of the
Enasic Equ.tbion . .... .34







NACA TM No. 1224
Page

6. CALCULATION OF THE COEFFICIEINTS OF THE SERIES DEVELOPMNTS
TI T-,r3 OF SPBERICAL ALD CYLINDRICAL FUNCTIONS 37
6.1 Continued Fraction Developments . 37
6.2 Method for Numerical Calculation of the Separation
Farameter and. the Development Coefficients .... 38
6.3 Power Series for Separation Parameter and
Development Coefficients.. ........ .... .39
6.4 Pover Series Developments ... .. .. 44

7. EIGEPFUNCTIONS OF TE BASIC EQUATION ............ 47
7.1 Limitation to v,lI Being Integers; v %-i. J 0 47
7.2 Breaking Off of the Series .. .. . .48
7.3 A Few Special Function Values . 50
7.4 Connection between the X- and Z-Functions .. 51
7.5 Normaalization eaaL Properties of Orthogonality of the
X-Functions of the First Kind .. .. .. 53
7.6 Generalization of F. F. Nevmann's Integral Relation 54
7.7 Zeros of the EigenfPnctions ... 55
7.8 Intesgral Equations for the Eigenfunctions .. 56

&. ASn PTOTICS OF TEE EIGENVALUES ANDD EIGLTiT'UCTIONS .... 57
8.1 Asymptotic Behavior of the Eigenvalues and
Eigenfunctions for Large V .. 57
8.2 Asymptotic Behavior of the Eigenvalues for Large
Real 7 .. .. . 59
8.3 Asymptotic Behavior of the Eigenfunctions for Large
Real 7 . 61
8.4 Asymptotic Behavior of the Eigenvalues for Large
Purely Imaginary 7 . .63
8.5 Asymptotic Behavior of the Eigenfunctiorn for Large
Purely Imaginary 7 .. ... .. .65

9- EIGENFUNCTIONS OF TEE WAVE EQUATION IN ROTATIONALLY
SYMMETRICAL ELLIPTIC COODINATES .. ..... 67
9.1 Larae's Wave Functions of the Prolate Ellipsoid of
Revolution .. . . 67
9.2 Lame's Wave Functions of the Oblate Ellipsoid of
Revolution . . 69
9.3 Normalization of Lame's Wave Functions for Outside
Space Problems .. .... 70
9.4 Development of Lame's Wave Functions in Terms of
Spherical and Cylindrical.F*unctions 72







;iACA TVl Hlo. 1224
Page

10. THE MEliHOD OF iGREET"'E F..2lSN FOP. THE SOLUTIOr Or BOUNDARY
VYALTE P;i 10.1 Green's FL.nction of the Wave Equation in .Radiation
Problems . . 7:
10.2 Developmexit of the Spnerical Wa-ve and of ;:r. .lane
Wave in Terms of Lam1's Ware .".;jc;,n.; .. 76
10.3 Diff'rction of a Scalar Sphlei ',-:. !':,ve or Plane
Wave on the Ellipsoid of Revolution .. 78

11. TABIES ... .. .. . . .. 80
11.1 C,-m'ier-ts to the Tables . .. .. 10
11.2 Eigonvaluas X (7) and Devolopmont Coeffi-
m
cionts a (7), b ,(7); Represonted. by Broken-
nor nr
Off Powor SSrils in 7 ............ 85
11.3 numerical Magnitude of the Eiornvalu-.)sa and the
DevcJlom ount Coefficients for a._i'orcnt n,7
and m = 0 . 5 95
11.4 Course of the Curv s X = An(y) or Lor Values
of th I x n . .. 102





































Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A.'Smathers Libraries with support from LYRASIS and the Sloan Foundation


http://www.archive.org/details/lameswavefunctio00unit







NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM NO. 1224


LAME'S WAVE FUNCTIONS OF THE ELLIPSOID OF PEVOLUTION*

By J. Meixner


1. INTRODUCTION


Lame's wave functions result by separation of the wave equation
in elliptic coordinates and by integration of the ordinary differen-
tial equations thus originating. They are a generalization of Lame's
potential functions which originate in the same manner from the
potential equation. Lame's wave functions are applied for boundary
value problems of the wave equation for regions of space bounded by
surfaces of a system of corfocal ellipsoids and hyperboloids.

For general elliptic coordinates Lame's wave functions have not
been fully calculated so far. Except for a few general properties,
not much is known about them. More consideration was given to Lame's
wave functions for the case of rotationally oymraetrical elliptic
coordinates (called for short, Lame's wave functions of the ellipsoid
of revolution). However, even for these functions few results are in
existence compared with those for the better known special functions
of mathematical physics, such as cylindrical and spherical functions.

The first more detailed investigation of Lame's wave functions
of the ellipsoid of revolution was made by Niven (reference 1) who
with their aid treated a heat-conduction problem in the ellipsoid
of revolution. However, the numerical values of the coefficients
of his series developments in terms of spherical and cylindrical
functions as they are given for the lowest indices contain several
errors which were taken over into the report by Strutt (reference 2).
A more extensive investigation with a greater number of applications
was made by Maclaurin (reference 3). M8glich (reference 4), whose
mathematical investigation of Lame's wave equation is based on
certain linear homogeneous integral equations, obtained results of a


*"Die Lameschen Wellenfunktionen des Drehellipsoids." Zentrale
fur wissenschaftliches Berichtswesen der Luftfahrtfcrschung des
Generalluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht
Nr. 1952, Juni 1944.







NACA TM No. 1224


more genera~ character. Strutt (reference 2) gives a survey of the
state of the theory of Lame's wave functions in 1932; he also demon-
strates on a large number of examples from acoustics, electrodynamics,
optics, wave mechanics, and theory of wave filters, the manifold
possibilities of application for these functions.

Of the treatises published in the meantime, an investigation by
Hanson (reference 5), which contains several new details, should be
mentioned, as well as a treatise by Morse (reference 6) on addition
theorems, that is, on the development of the plane wave and the
spherical wave in terms of Lame's wave functions, furthermore, a
number of treatises on the wave-mechanical treatment of the ion-
of the hydrogen molecule referencee 7). Kotani (reference 8)
deals with integral equations for Lame's wave functions. In
particular, a treatise by Chu and Stratton (reference 9) should be
pointed out which settles exhaustively the problem (treated so far
only incompletely) of the continuation of the solutions of
equation (2.4g) for large and small argument and shows in detail
how the entire theory of Mathieu's functions results as a special
and boundary case 'rom the general theory of Lame's wave functions.
Finally, a treatise by Bouwkamp (reference 10) on the theoretical
and numerical treatment of diffraction on a circular aperture is
to be mentioned which, for the first time, contains more detailed
numerical material-concerning Lame's wave functions of the ellipsoid
of revolution.

The main task of the present report on Lame's wave functions-
of the ellipsoid of revolution will be to compile their most
important properties in such a manner that. these functions take on
a form which facilitates their application. In this connection an
investigation of the solutions of the ordinary homogeneous linear
differential equations of the second order, which originate with
separation of the wave equation in rotationally symmc-trical elliptic
coordinates, is of importance; further, it has to be determined what
is to be understood in these solutions by functions of the first
and second kind, their normalization as well as the description of
the behavior of-these solutions in different domains of the
independent variables, in particular, their asymptotic behavior.
Here belongs also the indication of a method of numerical calculation
of these functions and the presentation of numerical tables.

For the purpose of clarity it was necessary to generalize
and supplement the existing material in some respects and to simplify
some of the calculations and proofs. Therewith the theory of Lane's
wave functions of the ellipsoid of revolution as a whole would seem
to have reached a development equivalent to the theory of Iathieu's











NACA TM No. 1224


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NACA TM No. 1224


2.2 Separation of the Wave Equation

Solutions of the wave equation in three dimensions are to be
determined. (k = wave number.)


Au + k2u = 0


(2.2)


of the form


u = fl()f2(T)f3(q)


(2.3)


Then the ordinary differential equations


a- [ t-





r l- 2 +

1 T .

d -.


-- +
1 -+ ,2


252



21 -
Yi- T12


k2c2c2 Xfl = 0


+ k c f2 = 0
- k2c2 2 )lf = 0o



k2 2 =


-k2c22 +X7)f2 = 0


-2,
"3 2f
^p =" T3


are valid for fl,


f2, and f3.


X and 12 are the separation
assumed to be any complex number'= .
for a given boundary value problem.


parameters. First, they are
Tipy can only be determined
In particular, g need not


(2.4a)


(2.5a)



(2.4g)



(2.5g)


(2.6)







NACA TM No. 1224


be an integer; this can be recognized, for instance, in the treat-
ment of an inside space problem in a sector 0 c = qcp of an
ellipsoid of revolution.


2.3 Reduction to a Differential Equation

The differential equation (2.4g) is designated as the basic
equation. (2.5g) is identical vith it;, (2.ha) is transformed into
it when 8 is replaced by tit and 1tc2 by -vIc'. Therewith
the investigation of the differential equations (2.4a), (2.5a),
and (2.5g) is reduced to that of the differential equation (2.4g).
The baeic domain, hocve:r, is not the same for all cases; it
extends from -1 to 1 in the cases (2 .a) and (2.5C), from ] to w
in the case (2.g), whereas the basic domain of the differential
equation (2.4a) in the transformation to (2.47) will be changed
to the domain from 0 to im (or .lse -im). It proves, therefore,
to be necessary to investigate the differential equation (2.4F)
in the entire complex E-plane.


2.4i Transformations of the Basic Equation

The basic equation represents a special cas;- of the linear
homogeneous differential equation of th-; second order with four
extra essential sing.ilaritiss, t.wo of which are made to join to
one essential singulsrity. The. latter is at infinity, the two
remaining extra essential singularities are at 1 and -1. The
present investigation of th.'- basic equation will start with
connecting its solutions with the solutions of limiting cases of
the basic equation. For k!c2 = O, the basic equation is trans-
formed into the differential equation o0 the sph-rical functions
and their associated functions or, as they iill be called here,
of the general spherical functions. If one lets the two singu-
larities at 1 and -1 combine into a single singularity at t = 0,
there originates, asic.e from an elementary transformation, the
differential equation of the cylindrical functions. This is
brought about by the substitution


S= 7'1 (2.7)

and the abbreviation


(2.8)


7 = kc







NACA TM No. 1224


if one then performs the limiting process 7,->0. From (2.4g) there
originates with


fl = (2 72)/2 -1 v() (2.9)


the differential equation


^2 2)d + 2 3 + .+ 2) d + 2 P. (4 + 1 v 0 o (2.10)


In the transition from (2.4g) to (2.4a) is transformed. into
itself and 72 need only be replaced by -72 For large distances,
that is, r2 = 2 + y2z2+ c2


r 1 r .4 2
S 22 O(2.1la)



2 k 2 + ,f2)lc2_
= kr2 1 + + 0 '+ (2.11g)



are valid.

Another important limiting case of the basic equation occurs
if, of the two singularities of the basic equation located at finite
distance, one or both move to infinity. Thenthe differential
equation of Laguerre's and Hermite's orthogonal functions, respec-
tively, is formed. This limitirni case will yield the asL.ymtotics
of the eigenvalues and. eigenfunctions for large absolute value of y.


2.5 Connection with Mathieu's Functions

Mathieu's functions are, in connection with Lam's wave functions,
obtained in two ways. They appear, as is well cnown, in the separa-
tion of the wave equation in the coordinates of the elliptic cylinder
and must, therefore, also appear in the limiting case of Lame's wave
functions for the ellipsoid with three axes when one axis becomes







NACA TM No. 1224


infinitely long. However, Methieu's differential equation is also
obtained, except for an elementary transformation, if p in (2.4g)
is set equal to t1/2. This also indicates that it is useful to
consider the basic equation not only for J that are integers,
but rather for arbitrary' coefficients V and i. The theory of
Mathieu's fmlctions is, therefore, a special case of the theory of
Lame's vave functions of the ellipsoid of revolution. Although
the present report does not yield ner results of Fathieu's functions,
it demonstrates hov they fit into a more general picture.


3. SPE-PIIC.';L iCmD CYLI'DRIiCA FUi'TIiC. S

3.1 A Few Formulas for Spherical Functions


The most important formaulas and theor.-ms for spherical and
cylindric functions needed belov are compiled and a few estimates
for these functions are given, which will be necessary for con-
siderations on uniform converglence of certain series in terms of
such functions. Magnus and Oberhettinrer (.reflernc,- 11) is again
referred to concerning the notation and -add.itional formulas. The
general spherical functions sa() and. _,(?) both satisfy the
differential equation


(1 I) () + |v(v + 1) 1 1 ) = C (3.1)
"v 1 2 [-V


and both satisfy the recursion formula


(2v + i P() (v )P+1 v + )P1( (3.2)



from which by three times repeated. application


. 2.pN_2 + + + 2 2 -
S= (2V + 1)(2V + 3) -v+2 + (2v 1)(2v + 3) -v


(+ [)(v -: l)p (1) (3.3)
(2v 1)(2v -,- 1) -V-2 _










NACA TM No. 1224


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NACA TM No. 1224


It is valid for -2v < arg z < 21( with the provision that the path
of integration for i < arg z2 < 2t and -2it < arg z < -it, respec-
tively, leads past the left of the point u = 0 and is also to be
returned there. The estimate has to be made as above. The maximum
of (u)r is, if the path of integration is suitably selected,
smaller than (Izl2 + )r for positive r's and smaller than br
for negative r's, where 8 is a number of the conditions Indicated
above, so that


(g.)^ (Z + L- r> )
V+r z! v


<= I -z
z r


for r = 0, 1, 2,

(3.8)

for r = -1, -2, -3, .


respectively, where a(5) and ( are positive, restricted
functions independent of r in each closed domain excluding the
points I = Tl, .


3,3 A Few Formulas for Cylindrical Functions

For the following it is more convenient to introduce not the
cylindrical functions themselves but rather the functions


(3.9)


They both satisfy the differential equation

They both satisfy the differential equation


2 -v v(v + 1)
d + d 2 v = O
( d


d2


. (3.10)







INCA TM Io. 1224


and the recursion fornxilas


V -2
L -- 2
2V 1 -2


2v + 1 f _1
t2 V 2v 1 v-2


2 + 1
+
(2V 1)(LV + 3)


2(2v 1) _
(2v 1)(2v + 3) v


V + 1 +2
V-, (3.11)
2V+ vI2



1
2V +


Besides, the eimpler recursion formula



(v r ) C+1) +V-(1)


(3.13)


is to be noted for the cylindrical
obtained by repeated application.
series development

00

4 ( ( ) =- 'v ,
2 2


functions from which (3.12) is
Finally, the consistently convergent


(3.14)


is given with


ar- t = 0


if t -is real and>0.


3.4 An Estimate for Cylindrical Functions


For th:" cylindrical functions Z
application of (3.13)


(Q) one obtains by repeated


2V +
t


r(p + 1)r v + p + .








12 NACA TM No. 1224






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NACA TM No. 1224


is in every closed domain, excluding the points 0 and o,
restricted (considered as a function of ); the upper limit dioes
not depend on r. If Re v < 1, this relation is valid at least
for such r's for which Re (v + r) > 2. A corresponding estimate
is valid for negative r's.


4. THE X-FUNCTIONS OF THE FIRST A2D SECOND KIND

4.1 Definition of the X-Functions of the First and Second Kind


Since for 7 = 0 the basic equation (2.4g) is joined to the
differential equation (3.1) of the general spherical functions, it
suggests itself to develop the solutions o' the basic equation in
terms of spherical functions. One formulates the two at first
formal series


X" r ()P(l) r(4.)
_-,
r=- o


2-)r a(l \ irU 2(7)Q ( (-2)
V Vr V-:-r
r=- oo


and attempts to determine the coefficients aL (7) and the index v
Vr
in such a manner that these t-.ro series formally satisfy the basic
equation and converge. The further problem will be to investigate
the convergence properties of the two. series (4.1) and (4.2) in
order to determine that, for the two series, one has to deal with
analytic functions which, in general, are linearly independent
solutions of the basic equation.

For the coefficients a (7) the indices V,p and the argument 7
v,r
will be omitted where there is no danger of confusion; the same
applies to the coefficients to be introduced later for series
developments of a similar hind.. The summation index r assumes only
even values. The term vith r = 0 in the two series (4.1) and (4.2)
is designated as the principal term of the series. In the solutions (4.1)
and (4.2) of the basic equation an arbitrary constant factor remains








NACA TM No. 1224


undetermined. It may be determined in some way. Then the series (4.1)
is denoted as X-function of the first kind.and the series (4.2) as
X-function of the second kind with the argument t and with the
indices V, with the parameter y. It will be found that the index v
is determined by the separation parameter X; more accurately,
there exists a functional relation between X, v, p. and 7
which is expressed by


x = (Y) (4.3)
V

The series (4.1) and (4.2) are now inserted in the basic
equation, the differential quotients of the spherical functions are
eliminated by means of the differential equation of the spherical
functions (3.1), and the factor E2 of the spherical functions is
eliminated by application of (3.3). Then there appears an infinite
sum of spherical functions with coefficients independent of t which
is equal to zero. The disappearance of the individual coefficient is
sufficient to this end. This leads to the conditional equations


21r = qrar-2 + Prar+2 (r = 0, f2, .
7


with the abbreviation


P 2(v + r+ 1)(v + r) 242 1
c = -x (v + r l)(V + r) + 72 (vr+ 1)(v- 2 -1
r (2V + 2r + 3)(2V + 2r 1)


(v+ r + + 2)(v+r +.1 (~4.
Pr (2v+ 2r + 5)(21 + 2r + 3)


S= (v + r i) (v + r 1)
(2V + 2r 1)(2V 2r 3)








NACA Tb No. 1224 15


4.2 General Oualities of the Coefficients a,

The recircion formula for the coefficients a., is int,.r-,reted
as a difference equation. In order to aroid .:omlic;tionr, the cace
of real fractional values of one-half for v is com:plct.cly e:xcluded.
and the case of real integ-,rs for v + I an. v -" rVe. ctively,
is postpone.!. Concerning the behai or of the coefficiets ay,. at
infinity, a simple formulation can be obtained ac.or'.inS to
iKeuser (reference 12). The equations


1im sup .r1 Ia 2[
r-oo r (4.!)

or


'%li cu .r'a : = rl (4.7)




are valid.

If the behavior of th' coeffici:-nts ar ?.t irfinity is given
by (4.6) at ler-t for nef-tlvee cr -pooitivc ri's, they inc.rase too
strongly to make a c-nvervt-enie 0o series (4.1) ..nr (4.2) possible.
Therefore, a solution of the difference equ..tion ((4.4) is to be found
which shows the behavior (4.7) faor r-4m ss c.ll CIa for r-.-r,
Althou.i th..re al--.s e:.ists a-n e:-:sct sol.it.ion W:ihicl behaves
for r-- -co as indizated in (4.7), this solution will in general
ehib-it for r---+-r4 th: behhav'or (' .6). 'Onrl;. fcor certain distinct
values of the paramiete-r v (r:-e so fir'). the blha.-:i ('.7)
prevail: for both r--m ar r--; in-:rce, in this manner
distinct values of -re coordinated to cs:h val~ e 0o v. For
y = 0 the c,-nr.ittions oare particvJ.rly sialul. Th-ere becomes for
all r's


(v + r )( r) a= (r = 0, .)


Th ns can, for a givan v, a.mrue -any of the value. (v + r + 1) x
(v + r). It is determined by the requirement that the series (4.1)
and (4.2) should be reduced to the principal terr' for this case,







NACA TM No. 1224


Trhich leads to X = v(v + 1). Now it is further required that
under (7) always the value should be understood ~ V
to v(v + 1) for y- 0. The existence of such a distinct X-value
to each given V, g, and e and its uniqueness will not be proved
here; it follows from the method of calculation given in section 6
for the determination of X.

From here on, the coefficients a (7) will always represent
V,r
that solution of the difference equation (4.4) Phich shows the
behavior (4.7) for r->t belongs to the value ~ (y), and
therefore has the boundary values


lim ar 0 (r = -2, t4, 6, .) (4.8)
7=0


Furthermore, the constant factor which is arbitrary in the coeffi-
cients ar may be determined in a given manner.


4.3 Convergence of the Series Developments of the X-Functions

of the First and Second Kind

From the estimates (3.6) and (3.8) as well as from the boundary
values (4.7) for r->+~o there follows immediately that the
series (4.1) and (4.2) in each closed .omacin, which does not include
the points = 1l, 0, will converge absolutr'ly and uniformly. One
may further conclude that the series (4.1) and (4.2) will converge
as well as the exponential series. Since the individual terms of
these series are analytic functions in this domain, there follows
from the uniform convergence that the.sums of the series themselves
will again be analytic functions, the singularities of vhich can
lie only at loo, 1.'rtherrore, that the series can be dif-
ferentiated. termwise, and therewith the fact that t-he functions
rerer:senter, by these series are real solutions of the basic equation.







HACA TM No. 1224


4.4 Further Solutions of the Basic Equation and Their Relation

to the X-Functions of the First end Second Kind

Between the general spherical functions P, P- F'
v v -v-I
P- Q, Q", Q Q' all of which satisfy, the same
-v-l -v v --v-1 --v-1
differential equation there exist, in general, six linear relations
independent of each other. They can be generalized for the X-furnctions
of the first and second. kinr. To this end several relations for the
coefficients ap (7) -rill be derived.
V,r

The system of equations (4.1) and the system of equations
originating from it by the substitution v-->-v-1 and r->-r are
identical because of

1I =bI q1'
-V-1,-r V,r -V-l.-r V,r



Due to the uniqueness of the solution there follows from it


(') % x1)' ((.9)
-v-1 V

Furthermore, the constant factor rhich is arbitrary in the ar's can
be determined in such a manner that


ai (7) = a (.) (4.10)
-v-1,-r v,r


The system of equations (4.4) end the system of equations
originating from it by the substitution s-->-i become identical
if one introduces in the latter instead of the a- 's the values
V,ri


b (7) = L(V + r ) (7) (4.11)
v,r r(v r p + 1) P(v + ; 1) V,r








NACA TM No. 1224


They are determined. so that bo = ao. Then the equations


b" (7) = a" (y)
v,r v,r

and

0 7(Y) = X1(Y)
V V


(4.12)


(4.13)


are valid.

After these preparations,at first a relation between X-( ( ).
V
X (1)(;7), and. 3x (y) is derived, since according to (4.1)

and (4.12)


CO

X-() (;Y) =
v _r=_
r=-oo


ib (7r)P ()
V 'r -'v+r


is valid.

If one expresses in this equation the spherical function P- (S)
v+r


by ( )
V+r(
and (4.11))



XV (1 ;)


and Q+r ()
V+r


the required relation



=Pv r+ l ) +.(1)
r(v + i +


(under consideration of (4.1), (4.2),


2 -e-pti
-
7C


sin it ) (2)( (4.15)


will be found at once.

In exactly the same way there result the formulas



-v-1 v


(4.16)













IIACA TN IJo. 1224 19





-- co 0 a .4


*- I-- ... f-




-ij 0 -; c/ .




r, -
,) -i II

--- 4 -C
,,, 4o, F i t'' ,', cr- 0 ,o




"'" Ti 0 > 1 l II
r- a -P c. J -.



S- r -- *, ii
rc'~i H r C

S.- o3 -P r- ,4
1. .P 0
.-i ..-i ci .-l .-.
XL > -- ,1 8
SC .. > -| I)

= I ,-- I' II









I II 1z C. -,-
ri r coi :! ,r-1 +
+ + O- > (j d +- i-.







4,t, ,-" -- ,D ,
r c ,
- -:_ *. K, ?, --, R.

L II C L'r C


C*ri -I *. .-- C *r- CT


5 I H-- 0 C
.- r > -- .;





. .) r 0 '
S- + ,- C, C O .-





C! i ., u"1, T U a' -. (r -




+ -0- ...- -- 1 4^ -. Q .
^. ,--.<-4 -- ,r r






C C
S0 -i N +





?Z! c, ED -. 1:1 >p
0* '. r -I I -
H o" r H 0 PO
N. i i -P C2- .l- r -P r, '

wflc -- a C' *-- H-:-
C- 0 '.-iP '. C. 0 C- N
'-- 3 r.i r-l )
?2 P ( -0 -P -P *'-


'2 H -. ,0 ;
r-F -. -2 r 4- *r-


4 ? H UH 02 C
r < 0 0 ..- 0 .-


,-O *- P0 r-i ..-H / N
F=0 E H 9 r/
5 Q i o c-i 0 -i
*'- W r-I U0 0 Pi 01










20 NACA TM No. 1224




11 Ci >



4P 4
d H*


I0
P ? >0






P +
+ 0 O O O

0 H o* *


0+ 0) 4) 1-0
i l H I+0 +)
u 41 CC\j

ed att

HH H Ii eH I < H



P O O 1+
-p E 'H 5H 5+P Q

Id.t-- en
-i H 0II
H cu 00 + m 0
0 C w a H2 0 *

Hpr CH C *S* *CMI
0 0 c 2 *I 2 CH ae
I = *>P 0 I r
0) 0 .- Pen 0
0) enH "3 i
H LA 0 'd


HI -*H *H 1+ c
en-P Hd a) r
H =L 0 >0 H



F ? *rH *H
0 ai *
en 0) b I
P- o |, s





-P *P
*r- s-- a d d .- I

en pe -
4-1 9

rd

H a H
O1 -H



0 2~- .


0 c h 4.?








NACA TM No. 1224


These tro general relations which are valid for I >1, are
in the ca"e 7 = 1 transformed not to exactly: (h.19) and (4.20);
eT1i, namely, represc.ntz in (i .23) end (4.24) an increase of the
argument by it, whereby under certain conditions the branch cut
may be passed, whereas the argument of -4 in (4.19) and (4.20) is
obtained by choosing such a path from E to -. that the branch
cut extending from -c to 1 iill not be passed.


5. THE Z-FUIrYCTIOIS OF THE FII'ST TO FOUrTH KITD

5.1 Definition of the ,-Functions of the First to Fourth Kind


If the tvo e'tra-,essential sLmnuilarities of thl- basic equa-
tion (2.1h) are made to join, as indicated in section 2, t]'ere
originates, aside from an elementary transforEnation, Beesel's
differential equation. Tt therefore suggests itself to attempt a
solution of the basic eql.atijon alo by seriess developm.nt2 in
terms of cylindrical functions. The functions .a"fined by the series
(which are at first formal)














series arc jcz
Substitution of these serir.s into the basic eqv.tion (2.h) (,t is
t to insert it into te transrd frm () (5 o














indices 4 and n by means of (3.10) and (3.11), and
V r Vr
r=- oo


are defined as E-functiro.ns of the first and. sconLd. kind. In these
sere2 2 '.
series ar < n; ar,- 2- = O, if arg c = arg r-.

Substitution of these aeri-s into the be.sic equation (2 .r) (it is
best to insert it into thle tr-inrsfori'ic-d form (210) of t. basic
equation), elimination of the fi-st ani Esecond ,ieriva-tiv.-s of the
indliccs and n by er-an of (-:.10) and (3.11), and
removal of the denomjnal.or 2 by means of (...12) lead.E finally,
exactly as in the X-fiunctions of +he first and second, kinrl., to a
three-term recursion system for the bi; (7). It a-rees vith the
v,r
recursion system (L.4) for the aP (7), if the 1 there is
v,r







NACA TM No. 1224


replaced by -.. The solution of the recursion system differs from
the indices bp (7) defined in (4.11) only by a constant factor;
V,r
this factor is selected to equal one. Therefore the relation (4.3)
formerly found between the index v of the generating functions
in (5.1) and (5.2) and the separation parameter X has to be
assumed also in this case.

As Z-functions of the third and fourth kind one defines


S3 ) (1) ;7) + i 2 (;) (5.3)



Z ( y) =Z V (1) ) 182 (5.4)


They have the same relation to Hankel's f'.nctions as the
Z-functions of the first and second kind to Bessel's and Neumann's
functions.


5.2 Convergence of the Series Developments of the Z-Functions

of the First and Second Kind

It must now be demonstrated that the series (5.1) and (5.2)
converge uniformly in a certain domain. One starts from the
estimate (3.17) and from the boundary values (4.7) which are also
valid for the b There results
v,r


lim sup) ri-7 < II- 1 (5-5)



The convergence is uniform. Thus the series (5.1) and (5.2) con-
verge uniformly and absolutely in the entire domain |{ > 1 with
the exclusion of the infinitely distant point; they represent







NACA TM No. 1224


therefore analytic functions, can be differentiated any number of
times termwise with respect to C, and satisfy the basic equation.
Only in special cases these series converge also for i1 1.


5.3 General Relations between the Z-Fumctions

The transition to various function branches over the branch
cut from -c through to 1 is made possible by the general relations.
They can be obtained corresponding to the case of the :'-fmnctions
from the general relations valid for the separate series terms,
thus for the cylindrical functions. (Compar- IMagnus and Oberhettirner,
elsewhere.) They read for | > 7(y that is, 15 > 1


1(l)(eir .yY) = evi Z(l)(t;y)
V %


(5.6)


Zp(2)(eti~ y) = e-,(v+l)fi ;4L(2)( t;)
V V


+ 2e-7na/2 sin (V +-)


cot (v (1) )
V


(5.7)


ZIL(")(e2'ti;Y) =









V


sin (7 l) V + -i
-e-7N/i2 2 t
sin (v +- & -ir

sin 7 v 1
- e-7i/2 e-(vl1/2)Ti 2.--. t(4)(y 7)
sin (v + )



-7i/,12 (v+1/2)i sin 7 +-(3) g .()
sin (v r+ v

sin (7 + 1) (v + In

sin (v + 2)it


(5.8)


(5.9)









24 IACA TM No. 1224



* H H H H
S"-- 0
S00
ca $4


SR)

--
a c, 0



gg k
0 0


o OJ-- a-
r 1
H Ho

I + + I
+a, a,






+ o


a, N |
0 I ,8

n Ig + rd v
H +0 H < a,



a, H rI, +0 P a 8
1 4z 0 0I~





S*ri +, o
a ) a,-P



IA + N, P -



(,0 ri uca
-PF fl +


-P a) 0 p'-
vi 0 C0 ->




*3 0* i bD
F+3 C\C d0
I S -








NACA TM No. 1224


This derivation is not accurate since the asymptotic develop-
ments (5.10) are further dependent upon the condition (i >> Iv + rl,
and this condition is not satisfied for all series terms of equa-
tions (5.1) and (5.2), respectively, since the sum has to be formed
over all r's from -c to o The fact that the developments (5.12)
are valid nevertheless is due to the behavior at infinity of the bar's
(compare equation (-.7)) according to which tha) series terms with
sufficiently large values of r do not contribute noticeably to
the Z-functions.

Equation (5.12) is proved as follows.

According to general theorems on the asymptotic behavior of
the solutions of homogeneous linear differential equations, the
coefficients of which are polynomials, one obtains asymptotic series
for the solutions by going into the differential equation (2.4g)
with a formulation of the form (5.12) and attempts to satisfy it
formally. This yields for the present case for the coefficients C(p)
the four-term recursion system




(p + )(p ) + (p + p + 7 (p) + +C(p ic 1) [P + l)p + 72 jC(p) + h ( + p)C(p 1)

+ 472(p + p)(p + p l)C(p 2) = 0

C(-l) = C(-2) = 0, p = 0, 1, 2, 3, (5.14)




from which they can be calculated recursively. This recursion
system, however, is satisfied just then when the series (5.13)
are substituted for the coefficients C(p). This substitution
leads after slight transformation to










MACA TM No. 1224


o
II




rl
54
-S
ri

+

4-
+


+







5--.
S4-







r-a










+
-4







--





5 -


a









m +
> o >









,
,r'





0 .



.,-I
ad 0co











al )
- P*h



q+ H



,-1
.o +
" 8





, r-t
0 + H



) ,rl --
, + p.
a) a


)*H H


'd rO -H


.H
to
03
*rl
p-P

** -5



0 o



* -
*rU) C












O
0 0
0


I
S o
H -H co



4-1 0 .








rl I
0 03










a) 0)
0 09
4- 0 4-


-P CU' (0
S.r- -I





O a 0
n P




.r0 a
*a) 3 0
+n> 0


H-> Pi




4 0 4


WH 0

3 e- 0)
m +3




n a


a)
ri 0)


ca

0 A


1 A -)
a)rd a a) P D
0 +*H 1 .
4t -1 4 -
a .- +



0 rdH4 o 0) HD
O --CJ A :
+* s -< 9 P





... D L-P w





,C +-' .I ,I 4-
CH 0 0)
d( 0 H +d




'.ri r O r) 0

A 0H O O -
-P k- a a
:* ap tiil d t)







L B 'r-I CD ) C


> a.r-4I' 0d 4-i
r0 M +0 pI
rd *ra- 0



a)H l*H ad ) O r P
a fl- 0 0 "0





H'--I 0VA *
+3a 0H.Ht O S0 o0)
Pd Cd ,+Q 4.p

a) 0)0 00 4-a
H. H ra) *H 1- 4 4 0

S0 i 'd t- 0 0a





Ms ( 4S3 B3 0 a 4*
i m- 0 P) 0


On O W *H '4-I 0 3
H 0 0 A, PHD
.r0 1 -1 +0







a) (- P St -I



) ) r + Od 0 0 0
(u r-i 03 (j w
* D E r- 4" +D O F*a 9-P








a) I0 t 0 0 a0 :
. a) a) 0d ar
pQ- T f Ofi O +> -
OEai -P ,t a 0 cir







0 0 t) l a O g l
k 0I O el
p r-I C C 0 0 0 + Da
o2p4 0 0 ,c| H





Se B (O t QB
k u do-, m


r-~~
.-





H
+







4-I
r
+
,-S








PI
l +
+ a
+ +





.,--




P-1


II







IMACA TM No. 1224


5.5 Further Solutions of the Basic Equation and Their Relation to
the Z-Functions of the First and Second Kind

It is immediately clear that with the Z-functions Z()

and Z p(2)(;7) the functions Zp(1'2 ) nd Zv(l'2)(;y) also
V-V-1 V
are solutions of the basic equation. Since there exist only two
linearily independent solutions of the basic equation, it must be
possible to express all solutions lincarily by two of them. Because
of the two relations


l() = -cos vnn (C) sin vai( ( -)
-v-v v L (5.16)

n-v-l(Q) = cos wvw( ) sin vnn () J


there follows irith the aid of equations (4.12) and (4.10) from the
definitions (5.1) and (5.2)


ZP(1)(L;y) = -sin vZW(1)(7) cos vZ4(2) (r;) (5.17)
-V-1 V V

Z() () ;y) = cos v7ri(1) ( ;y) sin vwZ' (tC ;) (5.18)
-v-l v v



In order to express the functions Z P1l2)(y) y (b 1 7)

and 2) ( 7)Y) it rill be practical to mve the asmptotic series;
it is sufficient to limit oneself to the first Lerm of the series.
Then there becomes



Z (3' ) =le-va I -- i_ b () 0(l -r)] (5.19)
r=-no








NACA TM No. 1224


The only difference for the asymptotic series for Z (34 7)

is that here al (7) takes the place of b1 (7). There follows
v,r v,r
immediately that
0o
(i;7) ) irb (7) = z(i) y) ira' (7)
Sr=- ,r r- ,r

(i = 1, 2) (5.20)

By combination of equations (5.17), (5.18), and (5.20) finally,
also the solutions Z-_!) ();7) of the basic equation can be reduced

to the two solutions ZP (1;r). As special cases of equations (5.17)
and (5.18)


ZN(3q) v) e~ vri ie(3)( ;7)
-v-1 v
(5.21)
1 (4) = -e i (4) ;7)
-v-1 v

should be noted.


5.6 Laurent-Developments for X- and Z-Functions

*The X- and Z-functions were introduced wholly independent of
each other. Since they all are, however, solutions of the same
differential equation, it must be possible to express, for instance,
the Z-functions of the first to fourth kind in general linearily by
the X-functions of the first and second kind. It will appear that
in general the Z-functions of the first and second kind are not
proportional to the X-functions of the first and the second kind,
respectively. Thus, it is not possible to define simply functions
of the first and second kind for the solutions of the basic equation;
it must always be added whether one is dealing with X- or Z-functions.










NACA TM No. 1224 29








0- + C
aj il t O 0 J '
S- + O +- COO -
S*rl >a *i h a

P C0 )o )' C)
II 0 r C H 4- +
*- O *f- -| P P O 0 W
P- 4-' H-' O -*H 'M 'P I
i-) .r -- 8 c- a 0 "
Pi 0 V a ) c 0 d p a +


0) + C0*
0>+ B 00 P 0(I no NO rt

> or-I P 0 mp ,i- 0- c
0 0 m f co -tl i H ) 02 4 z -C3
H rI .- T-I > Cd v ?
pa +2O0 ri 4 4 4 Id
i *r-Q *) T! r r rq
, go 0o; o ro -P Vfi
d i 4- o
ID p N i o- 0 0 +
- O D 4- oM

4 0 O4- O o 9 O H
4 ^^ aa r- *0 CD 14 0 I'Da 1Y
-P 0 O F-i H 4-0 r O P 0 |S
cO C- O 0- L O -r f.
4 O u- 0 '-I *H *u- 0 oI l.
-- ---t a a.u1 C > )
0 or-1 *0>C -O O -P 0 m 0 > s r-I
0% -m e O CM +
0 H aId M o 0 C3 4
b a0 a Iood ca nd m -P L *)i
433 0) 0 0-
Orl 9 mo mdH 4)
g I >3 O C
ro 0 ()>a 0e 4-.H C
O O Cik 0' '- a .1 (fl 1
rI0H O rd_ C a A
S0) 00)00 + 0
-rI o'd C a. *

cC N-, 0ii
4 ;- a), o a) 0 4-3 0L 0
p =L
( a)u 0 Oro o 0 i
I -r-1 I.p a *-i r- 4 0
mN O > I A m .43 a )
qH 0)O C 0u w00 C
S cr-4 c- 04 W M i 0 4 -4 ) 43


02 '4 *ri- %) ) A ( -
-)) O) G)/
!4 nin S- ) 0A
'0g 4'3 +3 4- +-
rI (3 m) su c. 1a c3 0 o

O 8 8N 0* i Ho-i -ori 11
O o: P .14 s o ( a a) r-i 14
+2 ) f rLC!3OO) CO C
0) 0 CO +2 1 -H P M CO a Q
O 0 I1 0 0 -H 0



A 8 1 4.n .- J ., C) C. -P C-
OO OO *H m r-4 .ri C3 (D
SI t- r-1 -r-1 p 0(U 0 0 -P 0)
RC r-O 1 H l M 0)024 PlC
oirN o a) 0 a+ 4 d -t
0) 0 '-- C,-CJ 0 CO -P r-I 5-0 fC r-H
-1dH *4 pH a *-
(1) LM 0 t> m +) m 0) E
r(D 0) ) a -
rd0)-iu Id a
S9 rd C) o- *H ID op d
o i- -r 4i 2o H a M .








30 NACA TM No. 1224




\ tr itr\







+ CC




;> It


*H g 0 C




r+ *H H



0i II'
8a o













Cm
Cl U)
=L P. H










411 +
8 lc
04 i
; j 8 81,xin 0









V0 2 a
II B






04'
V0



v v ^1
*)

Oa
F!1 8 cr:or r 8
mII r~c







NACA TM No. 1224


5.7 Connection between the X- and Z-Functions

Due to the equality of the characteristic e:cponents v
and -v 1, respectively, in equations (5.23) and (5.?4) or
equations (5.22) and (5.25), respectively, aX(2) ard IZ(1) on
V -v-I
the one hand and 1.(2) and S Z( on the otble only differ every
-v-i v
time by a factor independent of 6 Thus one may equate


(1 7) = 1 sin (v p)*e (y)()) (E (/) (5.26)
v 7 V -v-i

(1)(%+1)ii (2) L:
-z ( ) =e e () (t(;) (5.27)
-v-i V V


The various factors, as sin (v Pi, and co forth, were
introduced for convenience. Bctween K l (y) and 7(2)() there
V V
exists the connection

p(2)( 1 ) -(V+1)U! p(.)
K (7) in (v -)e (7) (5.28)
-v-1 = v

One further obtains with the aid of equations (4.18) and (5.17)

1)(1) r -yi. (1)
S( ;) = K (Y) (e coe vn) (C-)
v v L v

"- P+V i p 1 (2g) (5.29)



V V V V
vi z^2^c,;) + sin vn z c;) = eV7 () V(y)c)i2 () 7) (5.30)


Tf V, I are integers, these relations are essentially
simplified; then the X-fvnctions and the-Z-functions of the first
and the second kind., respectively, are actually proportional.







32 NACA TM No. 1224

Now the calculation of K (7) is left to be performed. To
that end one may select the coefficient of any power of E in
equations (6.22) and (5-25) and carry out the comparison. One
obtains

P(1) 1 v 2-2siv -.- )
S 7 =i pr &)


=- v,r
v,r




V,r
r=- ) (5.31)



+ + (1 +Sr



p(2) 1 -(v+P+l)it ")-2 s-2 is
S (Y) -e ) r(v + P + s + 1)


P irbP (7)
vr

V- 2 + 2 2
x (5-32)
Lira (y)






Any even number is to be substituted for s in equations (5-31)
and (5.32). The value of v(i)(7) is independent of the selected








TIJ.CA TM No. 1224


special value of s. If one replaces in equation (5.32) p by -p
and s by -s and then multiplies by equation (5.31),


r. (7) 4 (7) = e(+l) 7- (5.33)
V V

is originated.


5.8 Uronski's Detorminarnt

Wronski's determinant of the Z-functions of the firct and
second kind are defined by


V -Z(1) (;) .. ZP(2) (;y) g(2)( ) ()(;7) (5.34)
wz v d V v d- V


From the basic equation (2.4g) their follows in the Imnown
way that Wronski's '-etcrminant of any tTo of its solutions is
proportional to (~2 l)-1. The factor of proportionality is
determined by substituting their asymptotic series for the
-Z-functions of the first and second kind; it is sufficient to
limit oneself to the first terms (5.19). There results

m -2

S 11 r=- o


Wronski's determinant of the X-ftmctions of the first and
second kind


(2V()l) (t) ) (2) -tC A Y) (5.36)


results from Wg by using equation (5.30).







NACA TM No. 1224


(5.37)


is originated and therefore

W 1 1
7 t2 1


irb (

(Y) v (7)


Simplifications result for the important
First, one agrees upon omitting the index i
zero. No7r there is valid b (7) = a (7)
V,r V,r
to equation (4.1), because of _n,(1) = 1,


X(1)(1;Y) =
v;


(5.38)


special case p = 0.
when it has the value
and further, according


o00
r vr
v,r


(5.39)


Thus, one can also write for 'Wronski's two determinants


X (1) ..X (1) (l;7)
t -1


(5.40o)


5.9 Other Series Developments of the Solutions

of the Basic Equation

Niven (reference 1) investigated series developments of the
following form (the functions represented by them are called
V- and W-fuunctions):


( (7) = ( ,2 + 72)/ w-1
v


ric r(Y) (w)
Vr V+r
r=-> -


S.(r) (2) wx







NACA TM No. 1224


W4(1)(w;7) = ir a (7)$ (w) (5.42)
Vir v+r
r=-oo

The relation of the variable w to E ancd is:

2 = t2 Y2 = (E2 1) (5.43)

For the coefficients cr arid dr there results again a
three-term recursion system which can be transformed into
equation (4.1L). If the coefficient of the principal term is set
equal to ao, the equations


Ir + r + y +-
Cy>r(7) ( M
c (7) = a ,(7)
Vr r(VMrv + r j +vr



-=r --+r a (7) (5.44)


p (0) vde



-v1
() = all (7)
V,r pv + r _+ (V + g + v,r



= irt-V al (7) (5.45)
(o) v,r
-V


are valid.







NACA TM No. 1224


The series (5.41) and (5.42) converge uniformly in each closed
domain given by 17r < lvw < c; or, expressed in the t-plane:
1 < J2 < co. The bounding curve f,2 1 =1 is a
lemniscate. Equation (5.42) is, as will be shown later, a limiting
case of a general development, which still contains an arbitrary
parameter and which yields as a further limiting case the
series (5.1) and (5.2) of the Z-functions.

One can immediately give further series developments of
V- and W-functions; to this end one has to replace the
functions 'V+r in equations (5.41) and (5.42) by nv+r, or the
indices V + r by -V r 1, or g by -[i, or one has to make
two or three of these substitutions simultaneously. One thus
obtains a total of eight V-functions and eight W-functions. Their
properties will not be investigated here more closely; it should
only be mentioned that all of them also can be expressed linearily
by the --functions of the first and second k~ -d which is done in
the simplest way with the aid of the as ::_'c;i-ic series.

Whereas the asymptotic series of the .-f-.rctions progress with
powers of -1 the asjyra-ttic series of t] V- and. W-functions
one obtains from equation (5.41), and so fort'!, by substitution of
the asymptotic series of the cylindrical f-inc tons, contain powers
of (2 72) /, that is, 2 1)12. According to a
suggestion by Wilson (reference 7) one can now also set up asymptotic
series which progress with powers of ( t l)-l. They have compared
with the series (5.12) a slight -.vant:-ge in.: far as a three-term
recursion system results for their coefficients. Correspondingly,
for the solutions of the babic equation also developments in terms
of cylindrical functions with the argument u t 7 = 7(~ 1) may
be given, of the form



F1 = ( 1)/2( + 1)-1/2 e t ( 7) (5.46)
1 vt V+t


where t runs through all integers, the odd as well as the even
ones. +t can again be replaced by n and so forth.

These developments will, however, not be followed up here.







NACA TM No. 1224


6. CALCULATION OF THE COEFFICIENTS OF THE SERIES DEVELOPMENTS

IN TERMS OF SPHERICAL AVD CYLINDRICAL FUNCTIONS

6.1 Continued Fraction Developments


The solution of the recursion system (4.4) which for r-- -
has the behavior at infinity (4.7), can be represented by the
convergent (reference 13) continued fraction


ar = 2 r 9pr0r+2 r r/ r+2 I 4r+-29r+/ r+2 (6.1)
5r -2-r/$r ---- ^--- $r 1-j--/-%.+i (6.1)
ar-2 1 1 1

The solution which has the behavior (4.7) for r--> -m can be
represented by the convergent continued fraction


S Prd r r r-/W2 rr- 2 r-2r-4/-2r-4 (6.2)
ar+2 1 1 1


The subnumerators of both continued fractions are in each finite
closed domain of y- and X-values for sufficiently large values
of r in the case (6.1), of -r in the case (6.2) smaller than
one-fourth; thus, according to a theorem on uniform convergence
of continued fractions, the continued fractions (6.1) and (6.2),
respectively, are in each domain of this kind for sufficiently
large r's and -r's, respectively, uniformly convergent and
are therewith regular analytic functions in 7 and X, since the
individual approximation fractions are functions of this kind. For
not sufficiently large r's and -r's, respectively, then follows,
that these continued fractions are also analytic functions which,
however, need not in every case be regular.

A solution of the recursion system (4.4) has now to be found
which shows the behavior at infinity (4.7) for r--> o as well as
for r-> --. Then the value of ar/ar-2 calculated from
equation (6.1) must equal the value of this expression calculated







NACA TM No. 1224


from equation (6.2). An equation-redlts which for given V and g
allows calculation of the. separation parameter X as a function
of 7. If 7 and X V(V + 1) both are'sufficiently small, the
solution k (y) is a regular analytic function of 7 which
V
assumes for 7 = 0 the value v(v + 1). Thus the X'(7) as well
as the ar/ao can be developed in power series in 7 with non-
vanishing radius of convergence the magnitude of which will not be~
investigated here more closely.

*' :6.2 Method' for Numerical Calculation of the Separation- -

Parameter and the Development Coefficients

The representation of the coefficient ar by continued
fractions is also for larger values of 7 still particularly
suitable for the numerical calculation of the separation
parameter X and the are's. Mostly v, &, and .7 are given.
Then the values pr, qr, and r + X can be calculated
numerically from equation (4.5). One starts from a value for X
which is assumed as close as possible to the actual value and
calculates for a selected fixed r the expression ar+2/aZr
-from equation (6.1) as well as from equation (6.2). Then one
repeats this calculation with a slightly altered value of X and
exaiines whether thereby the agreement of the two values ar+2/ar
_is improved. By further variation of X one can finally obtain
.,. an agreement of arbitrary accuracy. Therewith one can find the
S lue X (7) with any desired accuracy.

One more investigation has to be made: whether-the solution
thus found for 7 = 0 goes over continuously into v(v + 1),
that is, into X () and not perhaps into X (0); for X (0)
Sv+2+2
also is a .solution of the present problem as can be recognized
from the fact that equations (6.1) and (6.2) contain the values v
and r only in the combination V + r. This question cannot be
decided unless one has already a general picture of the
functions X (7) as it is given in figure 1 for g = 0, v's that
are ntegers and el
are integers, and real y S.







NACA TM No. 1224


The number of terms of the continued fractions (6.1) and (6.2)
to be included in the calculation corresponds to the desired
accuracy. For large Irl the partial fractions 7 prqr+2/r r+2
assume the order of magnitude 7 /(16r4); thus the index r' of
the last partial fraction to be included will have to be selected
at any rate larger than j7j/2.

The calculation of the a (y) 's is made by taking the
v,r
value found for X(7) as a base, and calculating ar+2/ar from
equation (6.1) and therefrom a+2/ao, at+f/ao,


6.3 Power Series for Separation Parameter

and Development Coefficients

For the numerical calculation of the separation parameter and
the development coefficients one can for small values of 17l make
good use of the power series developments in terms of 7. If one
limits oneself in these to the first terms up to the fifth power
of 72, inclusive, one obtains, in general,still quite useful approxi-
mations up to about i72 = 5. Therefore, following, the power
series for the N (7) shall be calculated explicitly to 7 0
V 8
inclusive, for the ar/ao to y inclusive. Therewith one more
series term is obtained than by Niven (reference l); compared with
Niven's cumbersome treatment, the calculation is essentially
simplified.

For the limiting case 7 = 0 there follows from the recursion
system (4.4)

(7) + (v + r + 1)(v + r)ar = 0 (6.3)


The case where all ar disappear is not of interest since it leads
only to identically disappearing solutions of the basic equation.
Thus there becomes X (0) = v((v + 1), ar = 0 for r y 0.
V







NACA TM No. 1224


For y = 0 all 0r with the exception of 0o have nondisappearing
limiting values. From the continued fraction (6.1) one can draw
the conclusion


a = 71 +0
r-2r


(r = 2, 4, 6, .)


and from that further


a
r = Yr
wo-


(r = 2, 4, 6, .)


If one
well, there


takes the next partial fraction into consideration as
results as the next approximation


ar 7 ,"'^ 4 P "6
a0 7r J+ 7* +*.


+ +() (78)j


Accordingly,, one obtains


ar P-2P-4 Pr
ao -20-4 r

9rPr-2)
rr 2/ + o(78) .


(r = 2, 4, 6, .)


(6.6)


4(9-2P-4 9-4P-6
+ + .-4 +. .
\^2P1-4 i^-4 -6


(r = -2,-4,-6, .. )


(6.7)


One now substitutes.. a/ao, from equation (6.6) and a*2/ao
from equation (6.7) into the .equation .r= '0 of' the recursion
system (4.4) ,and obtains


(6.4)


(6.5)


q2 .14 .+ q ,








NACA TM No. 1224


^r2 1 c"P ^ -2 F, q -2p-41 R
74 0 + 2 41
V_


This equation permits the calculation of X (y)
10 V


as Over


(6.0)


series in terms of r up to the power 70, inclusive. At first
one can see, by having 7 approach 0, that 0o = 0(74). Tharewith,
however, 0r also is Iknown for any r with the exception of terms
of the second and of higher powers in y2. If ont now inserts c,
in'this approximation on the right side of equation (6.8), Do


becomes already; correct ji to the third power in
If one repeats this procedure with the new values
are correct up to 7y, inclusive, there results
therewith Y (7) exactly up to 710, inclusive.
of this calculation as well as the cflcu.lation of
not particularly difficult, therefore the results
timnediatuly.

In order to make the representation clearer,
abbreviations are introduced:


7", inclusive.
of the r which
finally 'o and
The performance
the ar's is
are giver.


the following


Or = Dr(1 725) + .


where


Dr = r(2v + r + 1)


5(= (2 1)
(2V + 3)(2v l)(P v + 2r + 3)(2v + 2r 1)


09D P2 -
D2 D4
D~ ".


(6.11)


-2q 0 P'-4 -2
DP,2 P- -4


(6.9)





(6.10)






NACA TM No. 1224


Ai = p22 + P -2 -2i


+ 8-
D -2


D-2


1 = 0, 1, 2, .


Then there becomes


2 2
Sv(v 1) + 2 2v + 2v 2,
Sv( + (2 + 3)(2 ( 1)


+ (ABoB A2 C)78 + [A 3- 2AoB,


+ p2?(2-2 + 7 10 + 0(712)
-2


Y2 12


- 72, + 746


+ 6Ao2 + Al
D2


Y4 9' 4 -2 ( 2


-A 4 y6
- AY + Al1


D2-
S1 -+(6 13 + )
(6.13)


P4 Ao
+ D2)

+ 4 2 + (710)


(6.14)


+ 5)


+ 7422 +


(6.15)


2 -A P Ao 1
82 64 2 A P6 1 0 + 0o(710)
8264 64 2 D 4 L)-


2
1 D2


C = 5
D2


(6.12)


X" (-)
V


P4 .








NACA TM No. 1224


a 7 6 6 72 52 + F64 6 + 0(7 1G)
aa q 2"



a8 q2q q68 + 0(y0)
a, D2D4gD8o


Y 2 P-2
-2 1
=7 -t 11


- 2 + 744. 2
- 5_82 5-2


. A.l

S-4) 5-2 + 0(710) (6.18)


212A, -2 + A


4 P-2P-4 '2(,_


+ 74 (_2 _25.-4 +


a:6
a r


a-8 '3 P-2P-4P-6P-8 10
ao 7 D.2D-DD8 (7
8, D_2 -4 -6D -8


o(y10)


(6.19)


(6.20)


(6.21)


For the case excluded. above where v has fractional values of
one-half, the convergence radii of these series equal zero. It seems
therefore probable that the convergence radii ere functions of v
which can be infinitely large for special cases, but not in general.


(6.16)




(6.17)


a-_2
a.


+ -4

S- A: P AG
6 L- D_ D 4



2- 2 -4 + -6) + o(7 1)


P-4
- --(25-2
D_2


,6 P-2P-4P-6
6 L







NACA TM No. 1224


6.4 Power Series Developments

Since occasionally power series developments of the solutions
of the basic equation (2.4g) also can be useful, they will be
briefly discussed below.

One can of course obtain them at once by substituting in the
series (4.1) and (4.2) for the X-functions of the first and second.
kind. the known power series develo-pmnts of the spherical functions
in terms of powers of ; one thus obtains power series for the
solutions of the basic equation which converge in the circle I0 < 1.
The problem of the Laurent-series for 1 < IS < need not be
discussed further since they are already calculated in equations (5.22)
to (5.25). However, one can obtain these developments directly.
Therewith a new method for the calculation of these functions and
particularly of X (7) is found.
V


One starts from the differential equation
written in terms of t rather than of .


(2.10) which is


g+ d[-~+ 2 v =


(6.22)


For the integration one tries the statement


v=-
s=-0


(6.23)


S ( s +s
gVi t


Then there results for the gs (the indices p and v as well as
the argument y in general are again omitted) the three-term
recursion system


(|2 -1) + 2
ar








NACA TM No. 1224


(v + s 4 + 2)(v + a s + l)gs+2

+ [(v + 3 + 1)(v ) X g 72g2 = 0

(s = 0, 2, t., .) (6.24)


gg *2
There exists a solution with the behavior at frnfinity -g >
gs-2 B2
gs
for s-)>= and a solution with the behavior at infinity ---- 1
gs-2
for s -a>-. The quotient of the two solutions in independent of s
only then when X assumes certain distinct values. As one can see
by comparing vith equation (5.23), these are just the values X (7).
V
From the behavior at infinity of the coefficients gs one can
conclude at once that the series (6.23) converges in the
domain 1 < |Il < .

If one substitutes in equation (6.24) for the coefficicnts gs
the coefficients calculated already in equation (5.23), there results
after elementary transformations


ira pl)
+rar (2) + r
+ ;.iaj;; + r- v
r=- v 2 2) L 2:73)

+ s 1)(2 + r a) v + r.+ 1)(v + r) -

Y2(v + s ) =

(s = 0, 2 t, .. .) (6.25)


These relations can be used, like equation (5.15), for the control of
numerically calculated values of the a (7).
v,r







HACA TM No. 1224


The recursion system (6.24) is, except for the case of v,IL
being integers with v k H 0, probably less suitable for the
numerical calculation of the X1 () than the continued fractions (6.1)
v
and (6.2).

Ordinary power series with increasing powers of result
for the solutions of the basic equation if one sets equal
g-2 = g-4 = = 0 and requires go = 0.

Then there results for v the determining equation


(v .g) (v 1) = o


(6.26)


Therefore v has here a meaning different from the one it had so
far. The behavior at infinity of the gs for s -> is simple:
all of them disappear. The behavior at infinity for s --- c is

given by g- ---1 or -->~. The first case is the standard
gs-2 s
case; the power series converges for }(g < 1. The second case is,
for V and beiig integers with v p (| 0, realized for a
solution of the basic equation, the X-function of the first kind;
the power series then converges for all finite .

It will be best to make the numerical calculation of the
coefficients of these power series which are convergent in the
unit circle so that first X (y) will be determined according
V
to the method given in section'6.1, or, for smaller values of 7,
from the series (6,13); the coefficients gs can then be calculated
from equation (6.24) for each of the two V -values given by
equation (6.26). A special but simple problem will then be left:
how the two calculated power series are connected with the X-functions
of the first and second kind.







NACA TM No. 1224


7. EIGFNFUNCTIONS OF TEE BASIC EQUATION

7.1 Limitation to v,p Being Integers; v ?= i| = 0


The determining factors for the eigonvalues of the separation
parameters X and p and, if occasion arises, of the wave coef-
ficient k, are the domain of space which was taken as a basis
and the boundary conditions on its boundary. This treatise is
limited to the most important type of eigenvalue problems of this
kind; for them the domain of space lies either within an ellipsoid
of revolution, or between two confocal ellipsoids of revolution, or
outside of an ellipsoid of revolution. The first t :o cases will be
called problems of inside space, the last case problem of outside
space. The entire domain -1 T= 1, 0 S cp 5 2g becomes then
effective for the two coordinates T and cp. Boundary conditions
in q and p do not ap-ear then; they are replaced by the require-
ment that the wave function for n = "l remains finite and that it
is single valued, that is, that it has the same value for cp + 2n
that it has for cp. The latter requirement leads to p's that are
integers, the first one to v's that are integers v > IPI 0.
That the X-functions of the first kind remain finite at the points
T = tl follows directly from the series (4.1) by teling the
estimate P ) + \2 112, which is valid for
this case, as a basis.

Following, n will always be Tlrtten for V and m for p
where v and p are real integers; for the present, n m 0 is
assumed. Ths case of negative m's, the absolute amount of which
is = n, is then obtained at once from equations (4.15), (4.17),
and (5.20).

The calculation of these special functions was practically
settled amongst other thinGs in the lact sections; even though it
was assumed there that neither v -- i nor v p are integers,
almost all results can nevertheless be taken over as simple limiting
processes demonstrate. Only a few particularities result, compared
with the general case; they will be discussed belov.








NACA TM No. 1224


7.2 Breaking Off of the Series


If gr, = 0 for a positive r' or pr, = 0 for a negative r',
the are's break off to the right or to the left, that is,


for q = 0;
r


for pr, = 0;


r' > 0:


r' < 0:


a = a =a = ,.
r' r'+2 r'+4


ar ar -2 r '-4


. = 0

= (7.1)
. = 0


is valid which follows in the simplest way from the continued
fraction developments (6.1) and (6.2), These cases occur when i -v
is a positive integer or when p + v is not a negative (Sicl) integer.
Since it was presumed 0 = m = n the first possibility does not occur,
but the second one does always occur, that is, for all admissible m,n.
Here again tro cases must be distinguished which are both oriGinated
from Pr' = 0:


n + m+ 2 = -r'> 0 or n+ m+ 1 = -r'> 0


In the first case m n is an even number, in the second,an
number; ar +2 is the first nonvanishing ar. For the b 's
follows from equation (4.12) that they disappear for all r


(7.2)


odd
there
r' + 2m.


Further, all pm (0) disappear for n + r = -m, -m + .m 1.
n+r
The developments of the X- and -functions of the first kind begin,
therefore, for n m = even with


i =(l)( ) =am im-npm( )
n n,m-n -m


+ am Im-n+2(m ,+
Sn,m-n+21 -m+2 +


()() = (2 im ,m(-n ) + bn,m-n+2mi.2( ) +* .








NACA TM ITo. 1224


and for n m = odd with


m(1) m m-n+1 m
X (1 7) = a i P (g)
n n,m-n+l Im+1

+ am imn-1r+3 m () +
n,m-n+3 -m+3

(7.4)

,m(l) =y ( a/2 -m kI
n= n- bnn+1 m- 1)

+ bti 3() '.
n ,m-n+3 n +3'



The series for iZ(1)(;vy) converges for all finite The
n
corresponding formulas for the -fiuctions of the second kind result
if the functions +r (t) are replaced by n (Q). The developments
for the X-functions of the- second kind show a special behavior. The
spherical functions of the second kind belonging to the vanishing
coefficients are, ar_-2, become infinitely, large in such a
manner that their products have finite limiting values. The
coefficients a (7) are defined by


Iin a () 0= a Pm (I )
v,r -v+r an,r -n+r

for m = 0, 1, 2, .. and v + r + m-> -1, -2, (7.5)


Then there becomes

m m+n-:- m
an,r ) = lim (-) r ( + v + m)P(-v-m)a (y)
Yvr


for m = 0, 1, 2, and v + r + m --- -1, -2, .


(7.6)







NACA TM No. 1224


and the series (4.2) reads

r' to
xm(2)' ;-T( i (n,r( irnr) Mra (7.T)
r=- oo =r +2


The series (5.41) for Vm(1)(w;y) breaks off only when n m
n
is an even number for odd values of n m the coefficients cm (7)
n,r
have the indefinite value o.0 if r -- r' with finite limiting
value. The series (5.42) for l(1)(v;y), on the other hand, breaks
off only when n m is an odd number; for even values of n m
the coefficients ed (Y) have the indefinite value o.0 with
n,r
finite limiting value if r = r'. Similar conditions exist for
the other V- and W-functions.

7.3 A Few Special Function Values

From the series (5.1) one obtains when arg (2 l) =
for = 0



m(l) im(r/2)m bm-n(7)l + for n even
-n (0) = (7.8)
0 for n m odd



m(1 0 for n m even
(l) =(7.9)
d "\im(7/2)m bm-n+i(7)/ ( + ( m) for n m odd
Vn-,m-,,,,,,k


50







NACA TM Io. 1224


The X- and g-functions of the first kind are for the
index values n and m considered here either even or
odd functions of t or 9, respectively, according to
whether n m is an even or odd value. Furthermore, because


of em(cos e + i.0) = (-i)mQm(co 9)
n n


X~(2)(0;7) = -i x(l)(0;
.:1 n (1)


- in pm(cos 3 + i.O) there is
2. n


for n m even


(7.10)


,m(2)
adx (0;7)
d


i (1)ml(07)
2i (or)


for n m odd


From WTronski's determinant (5.35) follows


= f br (b
[r=-co


for n m even (7.12)


a.m(1) (o;y)
n
6(


7m(2) (0;y)
n


G2
Srbm (y)
n,r
r=-Co


for n m odd


Therefrom the Z-function of the second kind and its derivative
with respect to ( for f = 0 can be calculated at once.


7 .4 Connection between the X- and -Z-Functions

If v,p are integers, considerable simplifications occur in
the relations (4.15) to (4.20), (4.24), (4.25), (5.6) to (5.9),
(5.17), and (5.18). They are so obvious that they need not be
discussed further. Equations (5.29) and (5.30) now assume the
simple form


(7.11)


(7.13)


7Zm(l) (0;7) n d(;)
ar







NACA TM No. 1224


-m(i) m(i) m(i)( )
-n n =)n (i)
n n n.


(1 = 1, 2)


For the m(i) (7) simpler expressions can be obtained if s
n
in equations (5-31) and (5.32) is selected in a suitable manner.
The same expressions, however, result in an even simpler way if
one substitutes in equation (7.14) and in the derivative of this
equation with respect to S, respectively, the special value 0 = 0.
m(i) m(i)
If one expresses () (O;7) and & (0;y)/di, respectively,
according to equations (7.8), (7.9), (7.12), (7.13), using
equation (5.13), there originates for n.- m = even


m(1) 1 1/2
n () =


m /2i bmm-n(7)
n2 )(o0) )r(;+ )


m(2) -1/2-m m-1 m(1)(O;l) m)
n \ m
a (7)
n,-n-m


P (7.15)


and for n m = odd


m(1)
n (7) =


1 1/2


m(2) -1/2
n


b m ()
n.m-n+1


(1)
n


-m(l) (O;)/ Q r( a m)
-m/ m-2 n 2 M
( a1 am ('7)
n,-n-im+l


(7.14)


S(7.16)


mi m1


(O,7 d) r( 5







NACA TM No. 1224


By m(l) (0;O7) and a '(l)(o;y)/d9 the valves of these
n n
functions are understood which result when E goes towards zero
from the positive imaginary half plane.
The distinction between mvea a~A odd n m can be avoided
if one sets, for instance, P equal zero in calculating the (i)(7)
n
from equations (5.31) and (5.32); the formulas (7.15) and (7.16),
on the other hand, have the advantage of greater simplicity.

7.5 Normalization and Properties of Orthogonality of the
X-Functions of the First Kind

The eigenvalues of the base equation Xm(y) are always real.
n
Proof of it is given in the known manner. Equally sJmply it can be
sho m that the functions X ()(;7) are orthogonal to each other,
that is,


I r
1


Xm(1) (;7)xml)C;7) cl = 0
n n


(7.17)


is valid for n / n'.
By inserting the series (4.1) into (7.17) one can also express
this property of orthogonality for even differences n n' thus:


a ()bm () 2 = 0o
n,r n',r+n-ny') 2n + 2r + 1


For the normalization integral one obtains


p1
-(I;y) (n +
1) 7) d = n-)'
f-1 r=--


for n i n'


(7.18)


2 m (y)bm (7) (7.19)
2n+2r+l n,r n,r








NACA TM No. 1224


7.6 Generalization of F. E. e1umann's Integral Relation


In the case m = 0
equation (2.4g) which is
the integral


F(4,)= 1 1
<-1


one obtains a second solution of the basic
independent of X(')(;7j) in the form of
n


(7.20)


e ) (y) dt
g- t n


The fact that this integral actually represents a solution
of equation (2.4g) is confirmed by substitution. The calculation
is reproduced in detail in Bouwkamp (reference 10). For large 5,
t and in the denominator of the integrand cancel in first
approximation and one can see then at once that J(t) is proportional
to the- -function of the fourth kind. The integral over t can then
be evaluated according to equations (8.20) and there originates,
because of equation (5.12),


J() = -i Z (7( ; l) (7;l (;7
n n / 4


iram ()
n,r


(7.21)


According to equations (7.8) and (5.33) the-Z-functions are now
converted to X-functions. Because of equation (5.39) there results
finally


1
'-1


ei (1)(t;y) dt
-t n.


(7.22)


Therefrom results for 7 = 0 F. E. Neumann's integral relation
between spherical functions of the first and second kind.


x(1) (t;)
n


X(2) (;) 17 ) (7








NACA TM No. 1224


7.7 Zeros of the Eigenfunctions

For m > 0 the zeros of the basic equaticn are situated
at I = 1, respectively, siace they there have the
behavior 2 l)m/2. If one divides the eigenfunctions by this
expression, the quotient does not have zeros at t = l1. In order
to understand this, one need only enter the basic equation (2.hg)
with the expression (t 1 I)m/2 multiplied by a power series
in (6 t 1). The zeros of (2 1)"m/2 Xm(1)(;y) are all simple;
n
if they were not simple, all higher derivatives of the eigenfunctions
would have to disappear there also. Since it is, however, a non-
identically vanishing analytic function, this case can never occur.
Further properties of the zerus of the eigenfunctions follow from a
simple consideration of continuity: namely, that in a nonsingular
point of the basic equation a zero cannot be newly originated for a
change of .* and an already existing one cannot vanish. There-
with the problem of the number of the zeros is essentially reduced
to the problem of the number of zeros of L6gendre'3 and their
associated polynomials and of Bessel's functions with on index of a
fractional value of one-half.

One deals first with the X(l) (;y) with real 9 and 7,
n
that is, with the eigenfunctions of the prolate ellipsoid of
revolution. All zeros are real; for this is valid for 7 = 0.
If, namely, for a change of y a complex zero would originate, the
conjugate-complex would originate along with it; but it contradicts
the simplicity of the zeros, that a real zero splits into two
complex zeros. The number of the zeros in the interval -1 < t < + 1
equals n m, that is, the number of zeros of Pm(g) in this
equan
interval. The zeros outside of this interval go over into the
zeros of Jn+I/2(Y) for 7-> O; the asymptotic distribution of
the zeros for large t is the same as'the distribution of Jn+1/2(7)
for arbitrary 7.

For the eigenfunctions of the oblate ellipsoid of revo-
lution X(m;iy) with real 9 and 7 also n m zeros are
n
situated in the interval -1 < 9 < 1; but now the remaining zeros



Translator's note: missing in the original.











NACA TM No. 1224


E- 0-=-
c./ cl
Oc Cu




0)
,* (D
t2 -0 r--



0 0 C ) 0
0H c PH -.' 0)
8 T g 0 -t 2a

*r-I rU O P



( + -H 0H

o)M 0) C L
r4 -P
S1H 1 0 )u


S 0 U .0 r u
N H ri 0
0' $4 I 'H + ) 0+ *ri


0 CHO
P4 0 f Q4 -P O Pr-

S4 0 p- r- + 0 4

a c ( *H PI oH
-r4 r a ,C *r + -O O fi, -H



0 ,2 M a ) ***I g O O S
0 k 0 2 k) 0- 0
riD 0 2 U o 0



0 k N -P -P
0 0 -Z 0 4-r rd rP
go a) a0 ,s O
*id 4 m 0 r -P I 0 S*


9 0 1 H H C i- -*
aH ) 0 *P- CH 0
05 t Di p Oq-4a 0) H S



4--* + 2 r 0
0 0 a -

0 H mr 0) 0 0 w M -H
o .m V- N -- H
OH '+-1> (> 0O
0 r 5 r0 0
0 *! f5 r m *o


01 0 O H ) H 9 -
C O a rP C?-'I 0 0
H O tan 0 a* r( O OI




-P k6 CH o,
*H P4 3 + l n I I P 0

'H Cd 0) A ? o

H- o s P ) 0
0) 0 S* *'*( < *rl 0 O
0) *r 0 p oG o o
0 +2

(D O O- O1 1 r ,t ,
0 0 )nd 0 0 )' ,M D -P H
(D Kih H ^ ^ t2+-D








NACA TM No. 1224


The integral equation (7.24) can be generalized. By selecting
another path of integration one can, for instance, also express the
Z-function of the second kind (that is, also the X-function of the
second kind) by an integral over the X-function of the first kind;
furthermore, equation (7.24) can be generalized to the case of
arbitrary v,p,7. However, the respective results shall not be
discussed here. Kotani (reference 8) indicated a general principle
for obtaining more general integral equations for the X- and
Z-functions, respectively.

Integral representations for the X- and-Z-functions have not
become known so far. It seems that the integral equations of the
type (7.24) or of another kind also can be substituted for them and
replace them; thus equation (7.24), for instance, represents a
very useful starting point for the investigation of the -asymptotics
of the X-functions of the first kind. The integral equation (7.24)
can perhaps also be applied when the values of the X-function of the
first kind are known only in the interval -1 < t < 1 and are to be
calculated for arbitrary real and complex E. (Compare the discussion
on the 7-asymptotics of the eigenfunctions in the following section.)
As M6glich (reference 4) has sho'wm, the integral equation (7.24) can
also be used for obtaining developments of the X-functions of the
first kind in terms of powers of 7.


8. ASYMPTOTICS OF THE EIGENVALUES AND EIGEIFUIECTIONS

8.1 Asymptotic Behavior of the Eigenvalues

and Eigenfunctions for Large v


The continued fractions (6.1) and (6.2) do not only have the
property to yield a development of X (7) in terms of powers of y
but in addition one can obtain from them a development in terms of
powers of v-1. It is more favorable to set up a development in
terms of powers of (2v + 1)-', because then the odd powers
of (2v+ 1) are eliminated because of equation (4.9). The cal-
culation itself is relatively simple so that the result can be
given immediately








NACA TM No. 1224


v(v -+ 1) + 2 + 1 4 162)2 + Y4
V8(2v+ 1)2

+ 1 (24 l6P2)Y2 + 46) +] 0 -(2V 1)
2(2v + 1)4 2


(8.1)


Presumably this series is not convergent but has asymptotic
character.

In order to form a judgment on the usefulness of the series (8.1)
for numerical purposes,, one gives for several cases the numerical
value of the remainder term denoted by 0 (2v + 1)-6J in comparison
with the value of the separation parameter X itself.

n 2 4 6 8

xo 0 11.7904 25.2513 47.10958 77.06246

Remainder term -0.134 0.0132 0.00095 0.00017


In a similar way one obtains the following expressions for the
development coefficients of the eigenfunctions for large values
of v.


a =
aO

a-2
a0


(8.2)


S- + 0 2V + 1)3
8(2v + 1) 2(2V + l)2


2
8(2v + 1)


2(2V 1)2 +


o(2v + 1) 3]


(8.3)


S --- + 0 (2
ao 32(2V + 1)2


+ 1)-3]


(8.4)


a4 -32(2V+ + 0 2 + 1) -3
a0 '32(2v+ 1)2








NACA TM No. 1224


8.2 Asymptotic Behavior of the Eigenvalues for Large Real 7

One limits oneself here to n and m vhich are integers
(n > m 0O) and to real large 7 vhich may be assumed to be positive
without essential restriction. Thus one obtains the asymptotics of
the eigenvalues and eigenfunctions for the wave equation in the
coordinates of the prolate ellipsoid of revolution.


An approximate picture
is obtained if one puts the
standard form


d-2
do2


+


of the eigenvalues and eigenfunctions
basic equation into tho Liouville


- 72 cos2 0 -


2 1
4
sin e


q = ( )(1- 2)1/4


S= cos 0


and interprets it as
interval 0 5 0 I


a Schridinger wave equation in the
of the potential energy (in suitable units)


n2 1
72 cos2 9 4 -.
sin2 4
sin 6


(8.6)


It has for large 7 at 0 = 2 a very narrow minimum and can there
very well be approximated by a parable. Then, however, there results
just Schridinger's vave equation of the harmonic oscillator, for
which eigenvalues and eigenfiunctions are knoim.

In order to obtain also higher approximations it suggests itself
to attempt a similar formulation as in equations (4.1) and (5.1).
One sets


F1 =( 2 -


1)m/2 "2rDT+r ( ,


S (8.5)


(8.7)








NACA TM No. 1224


equal, where the Dn are Hermite's orthogonal functions and the
functions of the parabolic cylinder, respectively. By substitution
of equation (8.7) into equation (2.4g) there results, if one utilizes
also the recursion formulas and the differential equation of Hermite's
orthogonal functions (see Magnus and Oberhettinger (referencr' 11)),
the five-term recursion system


4 l r2 + l4k + 87 + r + 1+ 4m2

2(N + r)2 2(N + r) 3r + W4(N + r + 2)(N + r + 1) p+2

+ (N + r + 4)(N + r + 3) (T + r + 2)(N + r + l)-r = O


(r = 0, t2, t4, ) (8.8)


The series (8.7) is probably not convergent; it rather represents

an asymptotic development in the sense that limits = 0 for all
o
even r f 0, or, as one concludes from that and from equation (8.8)


StL2r 2) = o(7-1-r/2)



190
.+(2 _0(_l-r) (8.9)


(r = 0, 2, 4, .. )


By a method of successive approximation the -r and X can be
represented as power series in 7-1. The calculation is elementary;
thus only the result is given. It is








NACA TM No. 1224


Xm(y) = (2N + 1)y 1(2 2+ 2N 3 4m2)
n 4

-(21r + 1) (2 N + 3 8r2)

+ -1 F-8m2(2e + 2N- + 1)
64y"
5(4 + 2N3 + + 7N + 3)] + 0("3) (8.10)


The connection between N, n, and m is given by counting the
zeros. For 7 --> w the de-7elopment (8.8) is reduced to the principal
term with the N real zeros of Eemrite's 1th polynomial, hereas
the X-function with the indices n and m to be Lapproximated has
exactly n m real zeros in the interval -1 < ? < 1. Therefrom
follows N = n m. For negative m on3 inserts instead N = n + m.


8.3 Asymptotic Behavior of the Eigenfunctions for Large Real 7

The asymptotic representation of the elgenfunctions results by
calculation of the coefficients 5r. They read, aside from the terms
of the order y->,








NACA TM No. 1224


- 25N 36)



+ 271 10ji (N 1)


+ 2 )


~2)3n(


- 1) (N 2)(N 3)


r- (8.u1)


"6
= -m
-o0 12872'


1- 2
S204872


- u
m N!
0 1287 2 ( 6)


-8
1 N!
S 2204872 (N-8).


According to the type of derivation, however, the eigenfunctions
are approximated by these series only in the interval -1 < t < 1.
In order to obtain an asymptotic series also for other t one starts
from the integral equation (7.24) and substitutes for the X-functions
in the integrand the series (8.8). Therewith the asymptotic develop-
ment of the eigenfunctions for all E is known; in particular, their
behavior can be investigated where, besides 7, 6 also is very
large. Since now the eigenfunction for all t is asymptotically
knownone obtains the solutions of the second kind by calculating the
integral in equation (7.24) with the asymptotic series of the eigen-
function and by means of another appropriate path of integration.


-.
327


a-2


= -1 [i
4y|_


+ -1(N2
32y


= 1+
327



32y L


'00


1 3N-+
27 L 2








NACA TM No. 1224


The zeros of the eigenfunctions located in the interval -1 < 5 < 1
crowd for large 7 more and more around t = 0; in order to understand
this, one has only to divide the zeros of Hermite's Nth polynomial
by \f and therewith to convert to the F-scale.

The domain of validity for equation (8.10) and (8.11) extends over
the indicated domain; thus originates, for instance, for m = tl from
equation (8.10) the asymptotic representation of the eigenvalues of
Mathieu's differential equation found by Ince (reference 14). However,
the limits for this domain of validity shall not be submitted to closer
investigation here.


8.4 Asymptotic Behavior of the Elgenvalues for

Large Purely Imaginary 7

One limits oneself again to n and m that are integers
(n = m = 0) and to.purely imaginary y of large absolute value.
This procedure yields the asymptotics of the eigenvalues and eigen-
functions for the coordinates of the oblate ellipsoid of revolution
(reference 10) and for the so-called inner equation for the separation
of the wave equation of the ion of the hydrogen molecule (reference 7).

The method applied in equation (8.2) fails here; yt2 namely
would become purely imaginary and the Dn(\ if27) would, for large t,
no longer decrease exponentially, but increase exponentially; they
would, therefore, be no longer appropriate for the development of
the eigenfunctions. The wave mechanical picture of the differential
equation (8.5) shows that in the case of purely imaginary 7 two
domains with low potential energy are present at 0 = 0 and 6 = 2n,
which are separated by a high potential peak with the maximum at e = .
One may, therefore, expect beforehand that the eigenvalues will
degenerate in first approximation; their split-up is exponentially
small in 7 I; it is the larger, the higher the eigenvalue. For each
eigenvalue there is an eigenfunction symmetric with respect to e = ,
that is, a = 0 arnan asymmetric eigenfunction.

The mathematical treatment is as follows. A singularity is made
to move to infinity. Then one obtains from equation (2.4g), aside
from an elementary transformation, the differential equation of









64 NACA TM No. 1224


Laguerrets orthogonal polynomials. This suggests for the solution
of equation (2.4g) the formulation of Svartholm (reference 7)


(1 e 2m/2 PE > t[ (m)
Fl(J) = 1 [ e [ W p(l -


(8.12)


wherein i7 was set equal to p; again it does not mean an essential
restriction if p > 0 is assumed. By substitution of equation (8.12)
into the differential equation 2,4g), application of Laguerre's
differential equation, and the recursion formulas for Laguerre's
polynomials (compare Magnus and Oberhettinger (reference 11)),-there
originates in the known way for the at a three-term recursion
system. With the abbreviations


X = -p2 + 2Tp (T2 + 1 m2) + A
2

T = 2N + m + 1


4At = (T + 2t 1)2 m2; Pt = 2t(T 2p + t)


(8.13)

(8.14)


(8.15)


the recursion system reads


at+l At+- + t-l A = (A + Pt)at


(t = -i, -N+l, -N+2, .


Therefrom follows for A the transcendent equation


A A2
A+ =


A22


S AO 2
A + P1


A-2
+P-2


from which A can be obtained as series in terms of powers of
Therefrom then.results


(8.16)


(8.17)


-1.







NACA TM No. 1224


m(7) = -P2 + 2rp (T2 + 1 m2) -(T2 + 1 2)


10r2 + 1 2M2(3r2 + 1) + m4]


64p2 [L

5---33T
512p


+ lll4T2 + 37- 2(23T2+ 25) + 13mj+ o(7 14)


8.5 Asymptotic Behavior of the Eigenfunctions
for Large Purely Imaginary 7
For the coefficients of the development (8.12)


0i T+T- 1 + +o -
1 1 1) +l T+2pJ+ 0171-)
an0- 16p 16p


2 (T
0o 5 L2p
a-1 i1
- = -


a-2


512


+ 1)2 (T + 3) m + 0(171 3)

1)2 -2 m + 9 +0( 7 |)


- 1)2 M [(T 3) 2 m2 + oy


(8.19)


is valid.


The significance of T
zeros. The principal term


and N results again from counting the
Sm)[-217(1- )] has N zeros which


for large jy1 lie all closely to t = 1. The real eigenfunction
has again N zeros in the neighborhood of = -1. For odd n m
another zero at E = 0 is added. The sum total of the zeros n m
equals, therefore, 2N for even n m and 2N + 1 for odd n m;
thus


(8.18)









NACA TM No. 1224


T=n+ 1 =2N + m + for n- m =even


T = n = 2N + m + 1 for n m = odd


is valid.

Baber and Hasse' (reference 7) calculated the series (8.18) with
the exception of the last two terms; only for the special case N = 0
they give also the last two terms; Bouwkamp (reference 10) calculated
the series (8.18) with the exception of the last term for the special
case m = 0. The asymptotic series (8.18) can still be used
for m = Vt; it then goes over, exactly like equation (8.10), into
the asymptotic series for the eigenvalues of Mathieu's functions
(reference 14).

For large values of ly7 the eigenvalues move closer and closer
together in pairs so that the asymptotic series (8.18) for the eigen-
values of each pair are the same (see equations (8.20)); that is,
the difference of the two eigenvalues has a stronger tendency to
vanish with increasing (7y than any power of 1/171.
(Compare table 11.)

The series (8.12) for the eigenfunction is useless in the
interval -1 : 5 0. There an approximation must be attempted
starting from the point 6 = -1. Since the eigenfunctidns become
exponentially small in the neighborhood of 0 = 0, one can build
up the eigenfunction in the entire interval -1 < < < 1 by combi-
nation of the two approximations starting from -1 and 1 and one
obtains


nm(1)p y) = Constant (1 2)2 co t 4I L2^p( M
t=-Go

Se- L [-2p(1 ) (8.21)
N+t

For even n m the positive, for odd n m the negative sign is
to be selected; in the one case the eigenfunction is symmetric, in
the other antisymmetric with respect to the point g = 0.








NACA TM No. 1224


'What was said in section 8.3 is valid for the asymptotic
calculation of the eigenfunctions and the functions of the second
kind. for any complex g as well as for the limits of the domain
of validity in the variables v,g,y of the asymptotic representations.

In order to show the use of the asymptotic series for numerical
purposes one compares for m = 0 a few eigenvalues with the values
resulting from equations (8.10) and (8.18) by giving the value of the
remainder term o(y73) and 0(p'4), respectively.


n 0 0 0 2

Y2 10 -25 -100 -100

.n(r) 2.305 -16.07904 -81.02794 -45.48967

Remainder term -0.025 -0.01616 -0.00008 -0.01528



9. EIGENFOICTIOI OF TEE WAVE EQUATION IN POTATIONALLY

SYMMETRICAL EIPTIC COORDINATES

9.1 Lame's Wave Functions of the Prolate Ellipsoid of Revolution


By separation of the wave equation in the coor-f.nates of the
prolate ellipsoid of revolution one obtains the following solutions
of the wave equation

u [= (1) k ) + Bg (2) (t;, [CX(1) (i )


+ MD (2) (I;YEeiPc + Fe-1n (9.1)
V v.


A, B, C, D, E, F are arbitrary constants, v and p arbitrary
real or complex parameters; the significance of 7 is given by
equation (2.8), thus 7 is real. The coordinates g and ( = 7A,
respectively, n and cp are real as well. Under X41,2) (7y) one
V








INACA TM No. 1224


understands %(1,2) ( + i X 07). According to the kind of the
boundary value problem presented, the arbitrariness concerning the
constants and parameters is limited; then such solutions of the wave
function have to be determined which remain finite for the entire
domain of the eigenfunctions.

Following, as before, three-dimensional domains only are dealt
with which lie inside or outside of an ellipsoid of revolution or
between two confocal ellipsoids of revolution. Then the domain of
the coordinates T and p is given by -i1 I 1, 0 c9 p 2A.
The requirement of single-valuedness and finiteness of the eigen-
functions then leads t? v = n, p = m, n Iml 0, and D =0.
The eigenfunctions are written in the form


U(e,1,pik5 = [Bnl)(P ) + m(2)(ti/nlc]ie1

(n = 0, 1, 2, .; m = t+, +2, .., tn) (9.2)


The domain of variables in. is denoted by 1 9 2
and g1 t 5 ~2, respectively., For the prolate ellipsoid of
revolution there is always 1 For inside space
problems 92 = finite, for outside space problems infinite. For
inside space problems boundary conditions for S1 and E2 are to
be prescribed. This results in two linear homogeneous determining
equations for A and B; they can be satisfied only for certain
distinct values of 7, that is, for certain eigenfrequencies; in
that case they fix the ratio A:B. In case El = 1 a boundary
condition can be prescribed only for. t2 > 1; -the boundary condition
for 5l = 1 is then replaced by the requirement of finiteness of the
eigenfunction at the singular point l l= 1; it leads to B = 0.

For outside space problems the boundary condition for E2 =
is eliminated; the functions (9.2) have for 2 -> co for arbitrary A
and B an oscillating behavior. One can see that immediately from
the asymptotic series (5.12). The boundary condition at 6 = 1 gives
the ratio A:B. For l 1 this boundary condition in turn is









NACA TM Io. 1224


eliminated and B becomes B = 0. A condition for the frequence
does not exist; all wave coefficients are admissible, the spectrum
is continuous and extends from k = 0 to k = -.


9.2 Lame's Wave Functions of the Oblate Ellipsoid of Revolution

The solutions of the wave equation originating by separation of
the wave equation in the coordinates of the oblate ellipsoid of
revolution are obtained from equation (9.1), by replacing 7 there
by tiy. Here also only the tIree-dimensional domains characterized
in section 9.1 are dealt with and the eigenfunctions can, therefore,
be written in the form


u (6,,cp;k) = A )(iy) + Bm(;i)(i)ei
n n n jn

(n = 0, 1, 2, ; m = 0, 1, 2, ,n) (9.3)



The domain of variables in T and (p is the same as in the coordinates
of the prolate ellipsoid of revolution. The domain of variables in f
is again denoted by 1 ( 2. For the oblate ellipsoid of
revolution there is 0 = What was said in section 9.1
for l > 1 is valid for inside and outside space problems
with l1 > 0. However, whereas there t, = 1 was a singular point
of the --function of the second kind, here t = 0 is a regular
point for all ;-functions. Thus, for determination of the eigenvalue
problem for Il = 0 in this case, also a boundary condition must be
given. The area (t = 0 is a circular disc. If such a circular
disc actually exists as a physical object, for instance, a circular
screen for problems of diffraction or a circular membrane, the
boundary condition on the disc results from the physical problem
taken as a basis. If, however, this circular disc has geometrical
significance only as singular surface of the coordinate system
taken as a basis, for instance, for the determination of the acoustic
or electrical natural oscillations inside an oblate ellipsoid of
revolution,the eigenfunction together with its derivative must be
required to be continuous at this circular disc which leads
to B = 0.











NACA TM No. 1224


4 -h

o -P



0 0
0









0w 0
o (ri 4

















0 -H 0
0 0
0 0 14







*4 o














a, -
0
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0 0 0



















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0 +3 4'









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r- oa






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Cu


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u











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tu

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5"


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0 ac

l r 'd- *
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+




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%.






*-t










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NACA TM No. 1224
01-


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02 +(


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ol
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4H .rl cr
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0










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+







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+







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*r






L



+





II

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i 3









NACA TM No. 1224


C0 O
H
01 *


B.
I:-
4a


0



Co*
0





H 04


a *S *


P014
r-1 1-*
04 Li S m





A43 0 0


ad







0 42H
-1 0rs


0 0d +3-r 0-4 :
em l



rd (a
0 S



0 m-p 2 +%g
a O CO
* e
















o 0 *'














r-,ro 01 m go
o 0 *

















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0 d M











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rt$ *


d
o






m
0
( 00
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c

$,4
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6 1
$4
+


$4
+
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at



+

of





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a































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o





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0



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*1
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i (D


k -







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8
U U
a












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31 D

-'
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--
'-f


*


v



,o
b-
y








NACA TM No. 1224


where r in the argument is given by r2 = x2 + y2 + z2 and must
not be confused with the index r which runs through all even
numbers. The development (9.9) is given already by Morse (reference 6).
One can interpret equation (9.10) as a development of the Z-functions
which contains still an arbitrary parameter r. For n-->l there
originates, if one divides before by (1 ,2)m/2, the series (5.1);
for nf->0 one obtains the series (5.42) for the W-functions of the
first kind. If one differentiates equation (9.10) with respect to I
and sets then n = 0, there results the series (5.41) of the
V-functions of the first kind. For ---> O there originates from
equation (9.10) the series (4.1) of the X-functions of the first kind.

At this point one can recognize why the formulations (4.1), (4.2),
(5.1), (5.2), (5.41), and (5.42) that is, the series developments
considered by Niven (reference l) all had to lead to the same
development coefficients ar.

Whereas equation (9.9) represents a development in terms of
eigenfunctions in polar coordinates which have their origin at the
point x = y = z = 0, Lame's wave functions can be developed also
'in terms of eigenfunctions in polar coordinates with the
origin x = y = 0, z = c. This development reads, as shown by a
simple calculation,


j-m(l) (C;r) (l ( ) em ()n+tE PM) (9.n1)
t=-WO


If one multiplies by (1 )-m/2 and then sets = 1, equation (9.11)
is transformed into the development (5.46). For g --- one obtains
a development in terms of spherical functions multiplied by sin (y7)
and cos (y7), respectively; the special case of this development m = 0
is already given by Hanson (reference 5).

If one finally develops Lame's wave functions in terms of the
eigenfunctions originating by separation of the wave equation in
cylindric coordinates, there results, with the aid of equation (7.24),









74 'NACA TM No. 1224





-A0 DH -

Cu O rd
Ol 0 a0



cu 0
O f M *o O
fq
(a110 r0d >0





0 4
(-" C 4-1 'o *3i



a COt) 0 -P 0 m
o Uo




-0 u IOD 0 t oH
4' ao
,o o 4- 1d i



*- -r 0q0 0 m we

S H O O pr 4d F
20,C -
,Dr-O 8 H -P 0

H tpo a .
19 0 0 0 H "0 P
N 8 a H H P P
0 o 4 HO
0O H

0 J --P PO D

6 H 0 OG 0 H ) i *rt o
0_0 1


24. P4 44- 1 p O 0 V 5
+, 0 AO
1 P 0 0 0 4a O)

m 0 0 d 0 qo- 0
'60 K "0 1
q-4 H +0 f 4P ooo

t N o o 9 "
go 0 -H 0^ 0 0
0oi $4r -H oq 0x P
0 o PP S 10 A -
d o >g^ m 0 |0
*0 M S w)

m -PFP 0 co
-H C8 H H

Vr Ot ;1
-P 0 4 P 0-P o 0 >0
H +- *P 'O -

O 0 4r O C M
43 do o

HOP O O rO

a, .i o: g i p- g
ip; E- r1^0S1-









NACA TM No. 1224


the right side of which is a (recently so-called) Dirac's 5 function.
It was introduced first by Sommerfeld (reference 15) and designated
by him as prong function. It has a singularity at the source
point Q in such a manner that


I 6(P,Q) dTp = 1 (10.2)


for each domain G which contains the source point Q, whereas
the integral has the value zero if the domain G does not contain
the source point. One can interpret 6(P,0) as limiting case
of a function which has for points of influence P in the neighbor-
hood of the source point Q a very steep prong whereas it decreases
toward the outside very rapidly to zero.

The solution of equation (10.1) is for outside space problems
uniquely determined only when besides the boundary conditions on
the bounding areas which are at a finite distance an additional
boundary condition at infinity is required, namely, the outgoing
radiation condition (or else the incoming radiation condition)
introduced by Sommerfeld (reference 15). According to this
condition, u(P,Q) for points P at very large distance from the
source point Q should behave like an outgoing (or incoming) wave.
One designates this solution because of its special properties as
Green's function G(P,Q k) of the wave equation pertaining to the
outgoing (or incoming) radiation condition. For physical reasons
the case of the incoming radiation condition will not be considered
below.

All developments of this section are performed for the coordinates
of the prolate ellipsoid of revolution; one obtains the corresponding
formulas for the coordinates of the oblate ellipsoid of revolution
by replacing 7 everywhere by iT.

Green's function can be developed in the following way in terms
of the eigenfunctions of the continuous spectrum



G(P,ojk) = k2 2 -m U p,cPp;) u ( nQcPQ7)27 (10.3)
Jo k2 K 2 n
n, m

The integration over i goes from 0 to o, the path of integration
deviating at the point i = k in the case of the outgoing radiation











NACA TM No. 1224


0'








Cu
8rt-





































II
it
v

OJ



a

9-








*-


4 0- 0

d 0




a P4S

rd -4 Pw

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a;n Rm



P4
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m ri 4 |
I m J





o 4 4.
S a 0


m 1-1 4!
Sm 9 ?4-i4

p.( 0 0(D
SV -ri

3 'n ? 4 )
0 oan -

) 0 0 rd
(k *ri I p

p4 m D

rd 0 oH s. m
0d a r

o? H kto N

a> 43a 0

4a -i .I D-




10 0 of

0 1 -. mo
k 0 ED
sq -1 :
(D o a m
0 it- N( 0 4-4
rd r >


0 0
od E


*ri Pi4- F

H '1



0 r

09*
0 o n .,-
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0
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a,) 0
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0* 0 0

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$4









NACA TM No. 1224


ICD
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4 ^p-



II

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0 0
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aH



mm
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NACA TM No. 1224


F +

4+> CH
mo

0

HaO
h
*
1114

0





*i









O
o
4'
SH
rd 4




*H 0
-1

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m
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H
+ -P


op 0




0 d4


:3 CH4 0


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P o -H
0 -H 54


o a0
0 0
C.
0 ++i 0








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o g ej
0 0,


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4 md
0 O -
0 -H -i


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r 'd P W
ed cil 3
e 0
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0 0 0 10
4' 0 10




















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0 od0
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CO 00p









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ca
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4-H 0 00


0


0'





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HC





.







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v












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a,
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p-4
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0
+


o


OU




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0










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r-1
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ALCA TM No. 1224


formed and the contention made that it solves the problem of dif-
fraction. Actually it represents a wave which comes as spherical
vave from the source point Q, satisfies on the diffracting surface
the given boundary condition (as does each single term of the sum),
and which behaves at infinity like an outgoing spherical wave. If
the source point Q, in particular, lies at infinity, Green's
functions represent the superposition of a plane wave and of an
outgoing spherical wave originating from the diffracting body with
an amplitude which, in general, is dependent on direction. Treat-
ment of the diffraction problem for a source point within a finite
region is omitted. One starts immediately from equation (10.8) and
contends that the solution of the diffraction problem of a plene wave
at the ellipsoid of revolution is given by


Sn m() ) ()


; n,m n Qi )7)


xm(1)() p;7)Xm(l)(n 1;7)eim(O-pCP) Im(y) 2 i-n ir b (7) (10.10)
r=-o


in the case of the boundary condition u = 0 for 5 = fl. Under m(7),
one understands therein the factor of normalization (9.8) with
AA* + BB = 1. In the case of the boundary condition = 0
for 5 = 5l one has to replace the two -Z-functions with tho
argument t1 in equation (10.10) by their derivatives with respect
to Cp at the point i'. The first term of the sum in the brackets
of equation (10.10) yields,when the sum over n,m is formed,exactly
the plane wave (10.8); the second term of the sum gives outgoing
spherical waves; furthermore, the wave equation and the boundary
(surface) condition are satisfied by each separate term of the sum;
the contention is therefore proved.

For the diffraction at the infinitely thin wire of finite length,
one has to set 1 = 1. -Z1m(3)(1;7) then becomes infinitely large
n
and, in equation (10.10), there remains only the plane wave. Thus
an infinitely thin wire does not present an obstacle for a plane wave.

















NACA TM No. 1224


For the diffraction at the infinitely thin circular disk, 7 is to
be replaced in the formulas by i7 aend 1 is to be set equal to zero.
(3)(0;iy) has a finite value so that the outgoing spherical waves
do not disappear; that is, even an infinitely thin disk represents
an essential disturbance for a plane wave striking it.


11. TABLES

11.1 Comnents to the Tables


The tables in section 11.2 contain power series developments
to y10, inclusive, for the eigenvalue Xm(7) according to
n


equation (6.13) and to y, inclusive, for the coefficients a (y
mn,r
and bm (7) according to equations (6.14) to (6.21) and
equation (4.11). Furthermore, to 7 inclusive, the coef-
ficients a la (7), according to equation (7.6), are given
nfrie an,o
for all those cases where a-2/aO, a.4/a0, and a_6/a0 disappear.
As far as the values of the coefficients ar/ao and br/bo are
not given in the tables, they disappear; then one must use for the
X-functions of the second kind the series (7.7) and the table for
the .c/ao.


The region of the n- and m-values in the tables extends
from m = 0, 1, 2, .. ., 9 and from n = m, m + 1, .., 9.
negative m, -which are integers, reference is made to the
relation (4.12).


)


For


The last given digit is, in general, probably certain; only where
the following digit after rounding up or off, respectively, is a 5,
the last given digit would have to be changed in a few cases by unity.
In the cases of the end digits ...5, ..50, .500, and so forth, it
is mostly indicated by a line over or under, respectively, the last
digit whether the respective decimal fraction had been originated by
rounding up or off.







N.CA TM No. 1224


For n = 0 the series begin to be useless only at 72 =10; for
larger n they can be used up to far larger values of 72. Below,
a few of the first eigenvalues for 72 = 10 are given as they
follow from the exact numerical calculation and from the poweri
series development to 710, inclusive.


n 0 2 4

XO(Vo) 2.305040 1.790395 25.25113
n

Xn F ) approximation 2.215 11.860 25.25147


Figure 1 gives a survey on the dependence of the lowest
eigenvalues on 7.

The tables in section 11.3 are taken from the thesis of
Bouwkamp (reference 10). They contain the eigenvalues X)(7) for
n
a number of pairs of values n, 72, and the coefficients am (7)
n,r
of the pertinent X-functions. These latter are fixed. so that


2n + 1 ar ( (ii.I)
2n + 2r + 1 _n,r (11.1)
r=-c


The integral of normalization then (compare equation (7.19)) has
the value

J X0(i)t; 7)12 2 (11.2)

Ln (11.2)n +
u-1

These tables contain further the values X(1)(l;7) and X(1)(0;7)
(1){ n n
for even and dX '(0;7) /d for odd n. The signs of the ar are
different from those of Bouwkamp since the present series (4.1)
and (4.2) contain in the coefficients a factor ir which is missing
in reference 10 by Bouwk.mp.









82 NACA TM No. 1224


Since the 72 assume in these tables only negative values,
these functions are appropriate for the treatment of problems
concerning the oblate ellipsoid of revolution or for the investi-
gation of the eigenvalues of the ion of the hydrogen molecule,
whereas, the tables in section 11.2 where 7 can be positive as
well as negative, may be used for problems of the oblate as well
as of the prolate ellipsoid of revolution.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics








NACA TM No. 1224


REFERENCES


1. Niven, C.: Philos. Trans. Roy, Soc. (London), vol. 171, 1880,
pp. 117-151.

2. Strutt, M. J. 0.: Lamesche, Mathieusche und verwandte Funktionen
in Physik und Technik, Ergebnisse der Mathematik und ihrer
Grenzgebiete, vol. 1, 1932, pp. 199-323.

3. Maclaurin, R.: Trans. Cambridge Philos. Soc., vol. 17, 1898,
pp. 41-108.

4. M'glishp F.: Ann. Physik (4), vol. 83, 1927, pp. 609-734.

5. Hanson, E. T.: Philos. Trans. Roy. Soc. (London), A, vol. 212,
1933, PP. 223-283.

6. Morse, P. M.: Proc. Nst. Acad. Sciences, vol. 21, 1935,
pp. 56-2.

7. Wilson, A. H.: Proc. Roy. Soc. (London), A, vol. 118, 1928,
pp. 617-635, 635-647.
Jaffs G.: Z. Physik, vol. 87, 1934, pp. 535-544.
Baber, W. G., and Hasse, d. R.: Proc. Cambridge Philos. Soc.,
vol. 31, 1935, pp. 564-581.
Svartholm, N.: Z. Physik, vol. 111, 1933, pp. 18.6194.

8. Kotani, M.: Proc. Phys.-Math. Soc. Japan, III, vol. 15, 1933,
PP. 30-57.

9. Chu, L., and Stratton, J. A.: Journ. Math. Physics, vol. 20,
1941, pp. 259-309. See also Stratton, J. A.: Proc. Nat.
Acad. Sciences, vol. 21, 1935, pp. 51-56, 316-321.

10., Bouwkamp, Ch. J.: Theoretischc en numerieke behandeling van de
buJging door en ronde opening. Diss. (Groningen).
Groningen-Batavia 1941.

11. Magnus, W., and Oberhettinger, F.: Formelrn und Sgtze fur die
speziellen Funktionen der mathematischen Physik. Berlin 1943.

12. Kreuser, P.: fber das Verhalten der Integrale homogener linearer
Differenzengleichungen im Unendlichen. Diss. (rubingen),
Borna-Leipzig 1914.







84 NACA TM No. 1224


13. Perron, 0.: Die Lehre von den Kettenbruchen, 2. Aufl.,
Leipzig und Berlin 1929.

14. Ince, E. L.: Proc. Edinburgh Boyal Soc., vol. 47, 1927,
pp. 294-301.

15. Sommerfeld, A.: Jahresber. d. Deutschen Math.-Ver., vol. 21, 1913.
pp. 309-353- See also Meixner, J.: Math. Zeitschr., vol. 36,
1933, pp. 677-707.








NACA TM No. 1224


m m m
11.2 Eignvaluea s (') and Develonment Coefficients a ,,(7), b r(7);

Repr-eenrterl tr, Brokern-Off Power Serilea in 7

ABLE i.- (r. x 1010 ..5 F, -. S1T 1r- y
I n I- t e


m = 0, n = 0 + '.:. 2
n = 1 +6000000000
n = *2 jt 78 ir- O,1 8
n 3 +5111111illl
n 4= +40649 1.506-
n = 5 +504273504L
n 6 +503030,020
n = 7 4502262.. L3
n : 8 *5017a1,--.860
n 9 +.011.005602

m = 1, n = 1 200000000C0,2
n = 2 +428571428
n : 3 +1666666661
n = 4, +9805194.805
n = 5 +4871794872
n = 6 +1.9009090o9
n =7 +4932126697
n = 8 +94 l7368421
n 9 +4 957983193

m = 2, an 2 +14-8571j29'2
n = i +33i33333333
n = +40?2597o40-6
n = +54.,897+4352.
n : 6 +4 4c.-5'L454
n = 7 b6606334u8%
n 8 +I 736842105
n = 9 +L769915966

m = 3, n = 3 +U 11111111-2
n = 1 +27?727?727
n = 5 + 504 27"=,50
n = 6 + 9 39 39 39 39
n =7 4+208144b796
n 8 +4 38596691
n = 9 +=.S0980 922

m = L, n = + 9'-0pgO090?-
n = = +?-307692-08
n 6 +3090909091
n 7 +3571660633
n = 8 +3894736842
n 9 +11176i.70..9

m = n 5 + 76920769Y2
n 6 +2000000000
n = 7 +2760180995
n 8 +3263157892
n = 9 +3611'45378

m = 6, n = 6 + 666666667v'
n = 7 +1764705882
n = 8 +2491228070
n = 9 +29971998880

m = 7, n = 7 + 58823529602
n = 8 +1578947368
n = 9 +2268907563

m = 8, n = 8 + 52631578972
n = 9 +1h485714'29

m = 9, n = 9 + 47619017672


-1E1%6 1916
- 68 .71' 2-:
S101 ,00918
+ 3. 1i0576

* 11298966
+ 78741 3L
* 581912?L
+ L 8:65 1
S-.62a0LO

- 4c.714 -86
- 8872'692
+ 13647587
* 11902117
4 8894604L
+ 6698782
+ .5174730
+ 6099403
+ 3320032

- 194 36 346,
- 2241h6689
- 2139-871
+ "'.84 89.
+ 3176672
+ -..6211
-, 300596
. 262083,

- 9976 306t1
- 13870427
- 1920040
- 877352
+ 755560
+ 1369050
+ 1958920

- 57793LLL'
- 9103-?:
- 8 '.9057
- 20 790 3
- 530540
+ ?l4418

- 36413297-
- 6274510
- 4157529
- 2301599
- 109649

- 2bo'00387y:'
_ 1499 41l
- 3460673
- 2221592

- 171 l0327"
- 3332L 31
- 2822595

- 12.96627
- 2535176

- 938951 k


+470') 6ib
- 6o.0- :1.
-476o91_
+ 5.9'-989
* 116'.
* 36,j
+ 1413
* 628
+ 310

+1219048 ,6
+ 1644~hO
-1182504
- 1315.03
- U166
- 10150
- 4008
- 1807
- 899

+ 36060076
+ 127901
- 309653
- 10857
- 39Q01
- 16771
- 7911
- o052

+ 132638v6
+ 76421
- 9'311
- 51-,68
- 27781
- 14. 3"
- 7813

+ 5731676
+ h ?60
- 28862
- 26567
- 17172
- 10467

+ 2788376
+ 26419
- 8311
- 12782
- 10172

* 1484076
+ 16310
- 1336
- 60o4

+ 847376
+ 104 35
+ 967

+ 511776
+ 6898

+ 33676


Sl-8356878
+ 25896
-i'.1089
- '1= 1.2


i5.04
S .3
+ 22
4 11

- 2103478
+ a,682
- 21357
- 5669

- 26
+ 6
+ 7
+ 2

- 5822Y8
+ 607
+ 6053
- 53
- 205
67
22
8

- 169878
- 83
+ 1759
+ 119
32
- 27
- 14

- 578v8
- 118
+ 570
+ 129
4 17
4

- 24.78
- 80
+ 205
4+ 79
+ 22

- 97,8
- 49
+ 80
4 44L

- 4678
- 29
+ 32

23r8
18

- 1378


-2L290.96T10
+ 872.80
+2 12.17
- 887.76
- 12.19
+ 1 .8,:
* 0.94
+ 0.15
4 0.03
+ 0.01

- 005.710 10
- 76.28
+ 229.7 '
* 78.30
- 2;.61
.9

0.08
0.02

4 57.71710
- 22.60
- 56.26
+ 23.27
1.22
- 0.58
0.30
0.07

+ 17.80710
- 4.51
- 18.21
+ .55
+ 0.53
- 0.01
- 0.04

+ 5.2 710
- 0.79
- 5.,5
+ 0.73
+ 0.29
-+ 0.06

+ 1.70710
- 0.07
- 1.82
+ 0,03
+ 0.10

+ 0.62710
+ 0.04
065
0.07

+ 0.22710
+ 0.0k
- 0.25

+ 0.11710
+ 0.,03

+ 0.02710


____ _~I___








NACA TM No. 1224


a (7)
m
TABE 2.- x 1010
Io


AS POWER SERIES I 7


m.= 0, = 0
n=1
n=2
n=3
n= 4
n =5
n=6
n= 7
n=8
n=9

=1, n= 1
n=2
n= 3
n=4
n=5
n=6
n= 7
n=8
n=9

m=2,n=2
n= 3
n=4
n=5
n=6
n=7
n=8
n=9

m 3, n=3
n=4
n=5
n=6
n=7
n=8
n=9

m=4,n= 4
n=5
n=6
n=7
n=8
n =9

m= 5, n = 5
n=6
n=7
n=8
n=9

m=6, n=6
n=7
n=8
n=9

m 7, n = 7
n =8
n= 9

m= 8, n= 8
n=9

m=9, n= 9


+111111111172
+ 400000000
+ 2.44897959
+ 176366843
+ 137741047
+ 112963959
+ 95726496
+ 83044983
+ 73325729
+ 65640291

+ 13333333372
+ 122448980
+ 105820106
+ 91827365
+ 80688542
+ 71794872
+ 64590542
+ 58660583
+ 53705693

+ 4081632772
+ 52910053
+ 55096419
+ 53792361
+ 51282051
+ 48442907
+ 45624898
+ 42964554

+ 1763668472
+ 27548209
+ 32275417
+ 34188034
+ 34602076
+ 34218674
+ 33416876

+ 918273672
+ 16137708
+ 20512821
+ 23068051
+ 24441910
+ 25062657

+ 537923672
+ 10256410
+ 13840830
+ 16294606
+ 17901898

+ 341880372
+ 6920415
+ 9776764
+ 11934598

+ 2306805672
+ 4888382
+ 7160759

+ 162946172
+ 3580380

+ 119346072


-3527336974
+ 3555556
+ 302904
+ 66996 6
+ 21683
+ 8738
+ 1071
+ 2105
+ 1177
+ 700

- 355555674
- 454356
- 120593
- 43366
- 18723
- 9160
- 4912
- 2826
- 1719

- 75726074
- 301482
- 130098
- 62411
- 32716
- 18420
- 10990
- 6877

- 2344867
- 151781
- 87376
- 50891
- 30700
- 19232
- 12481

- 9106874
- 78638
- 54963
- 36840
- 24727
- 16849

- 4119274
- 43185
- 34735
- 2590.5
- 18912

- 20793 4
- 25086
- 22451
- 18212

- 111037
- 15307
- 14900

- 667274
- 9743

- 41147/


-101900876
- 151062
+ 208818
+ 46389
+ 18710
+ 9467
+ 5462
+ 3439
+ 2306
+ 1622

+ 6134876
- 17898
+ 16210
+ 9870
+ 5930
+ 3767
+ 2524
+ 1768
+ 1285

+ 1222776
- 1430
+ 2386
+ 2541
+ 2057
+ 1579
+ 1207
+ 931

+ 300376
+ 165
+ 621
+ 828
+ 812
+ 718
+ 609

+ 91876
+ 210
+ 242
+ 329
+ 361
+ 353

+ 33276
+ 131
+ 118
+ 152
+ 178

+ 13676
+ 71
+ 64
+ 79

+ 6276
+
+ 37

+ 3076
+ 25

+ 1676


I ~







NACA TM No. 1224


TABLE 3.- anL x 101 AND x 1010 AS POWER SERiS
a otm ) a (()
1 noa0


m = 0, a 0

n 2
n 3
n34
n
n 6
n 7
n 8
n = 9

m 1, n 1
n =2
n 3
n -
n 5
n 7
3.8
n =9

* = 2, n 2
n 3
n =
n = 5
n 6
n 7
n.7
n 8
n 9

m 3, n = 3
n .
n 5
n.6
u 6
n 7
n 8
n 9

m 4, n 6
n =5
n =6
n =7
n 8
n =9

m = 5, n = 5
n =6
n =7
n 8
n =9

a = 6, n 6
n 7
n =8
n 9

S= 7, n 7
n =8
n 9

S= 8, n 8
n 9

S9, n = 9


+190676197"
+ 4535147
+ 2061l31l
+ 1177274
+ 760700
+ 531592.
+ 392250
*+ 30124=
+ 238563
+ 193571

+ 90702974
+ 687144
+ 506546
+ 380350
+ 295331
+ 235350
+ 191701
+ 159062
+ 131010

+ 1376297g
+ 168182
+ 163001
+ 167662
+ 130750
+ 115021
+ 101209
+ 89340

+ 33636716
+ 54'336
+ 63285
+ 65375
+ 63900
+ 60725
+ 56853

+ o186177
+ 21095
+ 28018
+ 31950
+ 33736
+ 36112

+ I421974
+ 9339
+ 13693
+ 16868
+ 18951

+ 186871
+ 4564
4 722a
+ 9175

+ 91374
+ 2110
+ 4061

+ 4824
+ 1354

+ 27171


-7.69601o6
+ 55817
+ 3740
+ 684
+ 189
+ 67
+ 28
+ 13
+ 7
+ 4

- 3349076
- 3740
- 879
- 28
- 111
5
24
13
7

- 374076
- 1466
- 608
- 277
- 138
- 73
- 42
- 25

- 68476
- 473
- 277
- 161
95
- p58
37

- 17076
- 166
- 124
- 86
- 58
- 40

- 527 6
- 61
58
46
32

1976
28
28
2

- 876
- 13
- 12

376
6

N6


+1 374-276
+ 24667
+ 8970
+ 426g
+ 2355
+ 1437
+ 940
+ 649
+ 466
+ 346

+ 352476
+ 2242
* 1421
+ 912
+ 65'
+ 170


* 208



+ 31L
+ 261

* 175
+ 143
+ 1l9
+ 11.


+ 5176
+ 79
+

+ 79
+ 72
+ 64

+ II,6
+ 22
+ 28
+ 32
+ 33
+ 32

+ 376
+ 7
+ 11
+ 13
+ 11
* I

+ 176
+ 3
+ 6
+ 6

+ 076
+ 1
+ 2

+ 076
+ 0

+ 076







NACA TM No. 1224


TABI I4.- -- x 1010
no(7)


AS POWER SERIE IN 7


m= 0, n= 2
n= 3
n=4
n=5
n = 6
n=7
n=8
n=9

a 1, a = 1
n=2
n=3
n=4
n 5
n =6
n 7
n=8
n=9

S= 2,n = 2
n=3
n=4
n=5
n=6
n=7
n=8
n=9

m = 3, n = 3
n=4
n=5
n=6
n=7
n=8
n=9

= 4, n=4
n=5
n=6
n=7
n=8
n=9
m = 5, n = 5
n=6
n=7

n=9

a=6, =6
n 7
n=8
= 9

m = 7, n = 7
n=8
n 9

a=8,n=8
n = 8
n=9

-9, n =9


- 22222222272
- 171428571
- 136054422
- 112233446
- 95359186
- 82840237
- 73202614
- 65561828

-33333333337
- 666666667
- 342857143
- 226757370
- 168350168
- 133502861
- 110453649
- 94117647
- 81952286

-133333333372
- 571428571
- 340136054
- 235690236
- 178003814
- 142011834
- 117647059
- 100163902

- 85714285772
- 476190476
- 314253648
- 228862047
- 177511793
- 14379085.
- 120196685

- 63492063572
- 404040404
- 286077559
- 216962522
- 172549020
- 142050628

- 50505050572
- 349650350,
- 260355030
- 203921569
- 165725733

- 41958042072
- 307692308
- 237908497
- 191222000

- 35897435972
- 274509804
- 218539428

- 31372549072
- 247678019

- 27863777172


+ 705467474
- 1523810
- 168280
- 42634
- 15011
- 6408
- 3113
- 1662

-133333333374
- 63492063
+ 9142857
+ 841400
+ 191852
+ 63047
+ 25630
+ 12009
+ 6232

- 63492063574
+ 76190476
+ 6310502
+ 1342964
+ 420316
+ 164766
+ 75054
+ 38087

+ 2666666677
+ 20614306
+ 4178111
+ 1260948
+ 480568
+ 214043
+ 106642

+ 49471433574
+ 9669343
+ 2837133
+ 1057249
+ 462332
+ 226858

+ 1899335274
+ 5419096
+ 1993669
+ 858611
+ 415905

+ 94451007
+ 3403334
+ 1446929
+ 693176

+ 5414397 4
+ 2276636
+ 1080271

+ 34024467
+ 1601021

+ 2281456Y7


+ 15197176
+ 42927
- 26152
- 8241
- 4452
- 2790
- 1889
- 1346

-54095238176
- 6478782
- 197018
+ 16156
- 7170
- 5138
- 3381
- 2295
- 1621

- 8055289076
+ 8760622
- 122277
- 4268
- 4295
- 3369
- 2449
- 1778

+ 8838009.76
- 2451288
- 64139
- 7860
3572
2475
1829

- 1295164876
- 380104
- 34429
- 6853
2950
1908

- 136583176
- 123724
19430
5180
2347

- 3701176
52456
11529
3762

- 12387876
- 25809
7154

- 5372076
- 14022

- 26503 6


__


---







ACA TM No. 1224



TABLE 5.- --x 1010 AND x 1010 AS PWl SI IE IN 7






a 4/ao x 1010 a^/a, x 101


m = 0, n = 4
n =5
n 6
n 7
n =8
n =9

* = 1, n = 3
n =
n =5
n 6
n 7
n 8
n 9

a 2, n 2
n 3
n = 14
n =
n 6
n=7
n 8
na8
n =9

a 3, n 3
n .
n=5
n 6
n 7
n 8
n 9

a 4 n = 4
n =5
n 6
n 7
n 8
n-8
n -9

m 5, n 5
n 7
n 8
n 9
n=9

S= 6, n = 6
n=8
n =9



n = 9

S= 8, = 8
n 9

m = 9, 9


+ 90702974
+ 687144
+ 504546_
+ 380350
+ 295331
+ 235350

* 19047619,r4
+ 4535147
+ 2061431
+ 1177274l
* 76070o
+ 531592
+ 392250

-666666667-'
* 95238095
+ 13605h.2
+ 481000o2
+ 23545148
+ 1369260
+ 885992
+ 616393

+28571I2867
+ 31716032
+ 9620010
+ 4238186
+ 2282100
+ 1392273
+ 921590

+ 634920637
+ 17316017
+ 7063643
+ 3586157
+ 2088109
+ 1335519

+ 288600297k
+ 11100011
+ 5379236
+ 3016591
+ 1869726

+ 166500177'
+ 7770008
+ 1223228
+ 2549627

+ 108780117
+ 57589147
+ 3399502

+ 7710238r'
+ 44145503

+ 5715647v


- 673176
+ 2610
+ 371
+ 106
+ 39
+ 17

+ 33862476
+ 100968
- 23492
- 2592
- 635
- 213
- 86

-12698t12776
+ 8165608
+ 1511520
- 274072t
- 259k4
- 5719
- 1776
- 677

* 5925925976
+ 8245723
- 127901
- 108971
- 22241
- 66514
- 2370

+ 2968460176
- 4144O0O
- 326912
- 62911
- 17587
- 6162

- 10853314L76
- 807274
- 148289
- 39919
- 13555

- 17 909376
- 309 39'
- 80722
- 26700

- 59066176
- 150108
- 18546

- 26280576
- 83016

- 13519776


- 160276
- 1207
- 852
- 610



- 1581076
- 11212
- 4830
- 2557
- 1524



+ 151171676
- 320667
- 48h49
- 14190
- 6392
- 3353

-3176603276
+10582011
- 1231668
- 134546
- 3622
- 11063
6705

+142328014276
- 384800Z
- 336361
- 79692
- 28127
- 12452

- 962001076
- 740001
- 159385
- 52235
- 21792

- 14o800176
- 296000
- 91412
- 36319

- 51800176
- 152353
- 58111

- 21476976
- 89808

- 13471276





NACA TM No. 1224


m /m
TABLE 6.- NUMERICAL VALUES OF THE COEFFICIE TS an,r ano

APPEARING IN THE SERIES DEVELOPMENTS (7.7)


-2 /ao x J10

m = 0, n = 0 +500000000072 -66666666674 +6613756676
n = 1 -1666666667 +222222222 -16825397


c-6/ao x o100

m 0, n = 0 +176366876
n= 1 -5291005
n = 2 +3703704
n = 3 529101
n = 4 50391
n= 5 7632

m = 1, n = 1 +265957476
n = 2 -3703704
n = 3 +1058201
n = 4 + 251953

m = 2, n = 2 +740740776
n= 3 -5291005







MACA TM No. 1224


[For


btm ()
TABLE 7.- -2-- x 10 AS PFOER SERIES IN 7
bm (T)
n,o

m = 0, there Is br = ar; compare therefore table 2a


=1, n 1


n = I*
n =2
n =3

n = 7
n =
n 7
1=8
n--9


m = 2, n
n
n
n
n
n
n
n

m = 3, n
n
n
n
n
n
n

m = n
n
n
n
n
n

m = 5, n
n
n
n
n


m = 6, n 6
n=7
n=8
n= 9


m = 7, n
n
n


m = 8, n = 8
n 9

a 9, n = 9


+8000000007 2
+k08163265
+264550261
+192837166
+150618612
+123076923
+103806228
+ 89620336
+ 78768349

+61224489872
+370370370
+?57116621
+193652501
+ 153866154
+126874279
+1075144403
+ 93089868

+49 382716072
+330578512
+242065627
+188034188
+152219132
+127097931
+10860o48'5

+41322314072
+295857988
+225641026
+179930796
+148280919
+125313283

+35502958672
+266666667
+209919262
+171093368
+143215181

+31111111172
+242214533
+195535278
+162310538

+27861660972
+221606648
+182599356

+24930747972
+204081633

+22675737072


-2133333371
- 1514520
- 301182
- 91068
- 34950
- 15704
- 78941
- 41357
- 2522

-113589037 4
- 2110372
- 607123
- 224681
- 98147
- 4821 3
- 2590S
- 14900

- 656560 374
- 1821369
- 655319
- 279902
- 135080
- 71* 32
- 4o0562

- 40980817
- 1441703
- 604588
- 287353
- 150013
- 81244

- 27186397,
- 1122807
- 526811
- 272002
- 151296

- 189213874
- 878023
- 449019
- 247677

- 13683477*
- 693939
- 379959

- 102089174
- 555324

- 78156774


+36808976
- 59660
+ *0526
+ 20727
+ 11069
+ 6158
+ 4057
+ 2702
+ 1885

+18340476
- 10012
+ 11136
+ 9149
+ 6171
+ 4135
+ 2844
+ 2017

+ 8107176
+ 1978
+ 4651
+ 1553
+ 357*
+ 2666
+ 1980

+ 4129376
+ 3851
+ 2666
+ 2567
+ 2193
+ 1766

+ 21880,6
+ 3110
+ 1790
+ 1601
+ 1422

+ 1238576
+ 2623
+ 1285
+ 1072

+ 741076
+ 1939
+ 952

+ 464* 376
+ 1422

+ 302576







NACA TM No. 1224


TABLE 8.- -4) x 1010
bm (7)
nso


ABD x 1010 AS PO SfRIES II
bm ()
n,o


a = 0, there is br = ar; compare therefore table


m = 1, a = 1
n=2
n= 3
n =4
n 5
n = 6
n 7
n 8
n 9

S= 2, = 2
n 3
n =4
n=5
n 6
n 7
n 8
n 9

m = 3, n = 3


n =
n =
n 9


n=8
a = 4, n = 4
n = 5
n =6
n =7
n =8
n =9

m = 5, n = 5
n = 6
n =7
n=8
n =9

m = 6, n = 6
n = 7
n=8
n =9

m = 7, n= 7
n = 8
n 9

S= 8, n = 8
n = 9

m = 9, n = 9


b4\ x 1010 b6/b x 101


+1360544274
+ 4810001
+ 2354548
+ 1369260
+ 885992
+ 616393
+ 451867
+ 344592
+ 270999

+ 96200107-
+ 4238186
+ 2282100
+ 1392273
+ 924590
+ 652697
+ 482428
+ 3695144

+ 7063643~7
+ 3586157
+ 2088409
+ 1335519
+ 913776
+ 657857
+ 492725

+ 53792367
+ 3016591
+ 1869726
+ 1246058
+ 877142
+ 644333

+ 422322874
1 2549627
+ 166111i
+ 1147032
+ 828428

+ 33995027y
+ 2172614
+ 1474756
+ 1049342

+ 279336174
+ 1868024
+ 1311678

+ 23350307,'
+ 1620308

+ 19803767


-50235176
- 26177
- 4104
- 1021
- 333
- 130
- 58
- 28
- 12

-26176976
- 36934
- 8508
- 2615
- 97m
- 417
- 198
- 102

-14363376
- 31198
- 9154
- 32k
- 1362
- 630
- 317

- 8423376
- 23800
- 8277
- 3343
- 1512
- 747

- 5235976
- 17737
- 700a
- 3107
- 1509

- 3415976
- 13231
- 5770
- 2761

- 2319776
- 9966
- 4706

- 1629076
-7602

- 1176976


+9866776
+26909
+ 10654
+ 5181
+ 2874
+ 1746
+ 1135
+ 777
+ 554

+6727376
+23'39
+10362
+ 5337
+ 3056
+ 1892
+ 1243
+ 856

+4687876
+19241
+ 9339
+ 5094
+ 3027
+ 1921
+ 1284

+3367876
+15565
+ 8150
+ 4678
+ 2881
+ 1876

+2h90y76
-+12595
+ 7017
+ 4211
+ 2680

+1889276
+10256
+ 6016
+ 3752

+1465176
+ 8423
+ 5160

+1158176
+ 6981

+ 930876


I __ __ _






NACA TM No. 1224


TABLE 9.-


bm
n-2L x 1010 AS POWER SERIES IN 7
bm
n,o


m = 0, there is


br = ar; compare therefore table 4]


m = 1, n
n
n
n
n
n
n

m = 2, n
n
n
n
n
n

m = 3, n
n
n
n
n

m = 4, n
n
n
n

m = 5, n
n
n


m=6, n= 8
n=9

m = 7 n= 9


-5714285772
-68027211
-67340067
-63572791
-59171598
-54901961
-50992534

-2267573772
-33670034
-38143674
-39447732
-39215686
-38244400

-1122334272
-19071837
-23668639
-26143791
-27317428

- 635727972
-11834320
-15686272
-18211619

- 394477372
- 7843137
-10926971

- 261437972
- 5463486

- 182116272


+1523810 4
+ 252420
+ 76741
+ 30022
+ 13730
+ 7005
+ 3878

+ 42070074
+ 191852
+ 90068
+ 45768
+ 25018
+ 14542

+ 14921874
+ 105079
+ 64076
+ 38917
+ 24237

+ 630477
+ 57668
+ 42030
+ 29084

+ 3020774
+ 33024
+ 27422

+ 159007Y
+ 19805

+ 900274


n = m + 1, b-2/bo disappears.


[For


-3283676
+ 4847
- 2868
- 2447
- 1811
- 1339
- 1009

- 815276
- 610
- 920
- 936
- 816
- 679

- 229176
- 655
- 476
- 455
- 416

-76576
- 374
- 268
- 242

- 29476
- 199
- 152

- 12776
- 107

- 6076


_ I__~C I


For n = m and






NACA TM No. 1224


TABLE 10.- X 1010
bm (7)
n,o


AND na-6 x 1010 AS
bm (7)
n,o-


POWER SERIES IN 7


For m = 0, there is br = as; compare therefore table 5]



b ob x 1010 b4b0o x 1010


m = 1, n = 5 +13742974 -156676
n = 6 +168182 371
n = 7 +163007 136 -17276
n = 8 +14766~ 59 -213
n = 9 +130750 29 -203

m = 2, n = 6 + 3363674 -37176
n = 7 + 54336 -227 6
n = 8 + 63285 -127 307
n = 9 + 65375 72 51

m = 3, n = 7 + 1086774 -10676
n 8 + 21095 99 6
n = 9 + 28018 72 77

m = 4, n = 8 + 421974 3676
n = 9 + 9339 43

m = 5, n =9 + 186874 1476


For n = m, n = m+ 1, n = m+ 2, n = m+ 3,


b-4/bo disappears.
For n = m, n = m+ n = m+2,
n = m + 5, b_/bo disappears.


n = m+ 3, n = m+ 4,