' I a

Reproduced

armed Services Technical Information figencyARLINGTON HALL STATION; ARLINGTON 12 VIRGINIA

NOTICE: WHEN GOVERNMENT OR OTHER DRAWINGS, SPECIFICATIONS OROTHER DATA ARE USED FOR ANY PURPOSE OTHER THAN IN CONNECTIONWITH A DEFINITELY RELATED GOVERNMENT PROCUREMENT OPERATION,THE U. S. GOVERNMENT THEREBY INCURS NO RESPONSIBILITY. NOR ANYOBLIGATION WHATSOEVER; AND THE FACT THAT THE GOVERNMENT MAYHAVE FORMULATED, FURNISHED, OR IN ANY WAY SUPPLIED THE SAIDDRAWINGS, SPECIFICATIONS, OR OTHER DATA IS NOT TO BE REGARDED BYIMPLICATION OR OTHERWISE AS IN ANY MANNER LICENSING THE HOLDEROR ANY OTHER PERSON OR CORPORATION, OR CONVEYING ANY RIGHTS ORPERMISSION TO MANUFACTURE, USE OR SELL ANY PATENTED INVENTIONTHAT MAY IN ANY WAY BE RELATED THERETO.

5N'SSFE

ii

'II

L _-_

APOP-

THERM ADVANCED RESEARCH

H A rA E w Y 0R K

EXPRESSION AS A LEGENDRE FUNCTION, OFAN ELLIPTIC INTEGRAL

OCCURRING IN WING THEORY

by

B. 0. U. Sonnerup

TAR-TN 59-! November 1959

Submitted to the Air Branch, Office ofNaval Research in Partial Fulfillment of

Contract Nonr-2859(00)

Donald Earl OrdwayHead, Aerophysics Section

Approved: OKA. Ritter

DirectorTherm Advanced Research

Reproduction in whole or in part is uermittedfor any purpose of the United States Government

ABS -IRACT

An integral Lha> often arises in potential

theory is shown to he related to half order Legendre

functions of the second kind. iric!uding a special forn

of the integral studied by Riegels. Series expansions

of these Legendre functions for values of the argu-

ment in the neighborhood of unity and much greater

than unity are also presented. For convenient

reference, recursion formulas and expressions in

terms of elliptic intearals are noted-

iI

SYMBOL:Z

A,BC constants

a real constant

ans•bns coefficients in expansion of Qn-X.

E(k) complete elliptic integral of first kind

Cn(k ) Riegels' function

J (z) Bessel function of first kind and order v

K(k) complete elliptic integral of first kind

k modulus of elliptic integrals

m real integer nutiber

n real integer number

Pv(Z) Legendre function of first kind

Qv(z) Legendre function of second kind

R spatial distance between singularity andfield point

(x,r,&) cylindrical coordinates

U: (n) Gauss' function II n) = ['(n+l)

v complex number

iii

I, I NTRODUCTI ON

The method of supcrrmpos-_riq or dastributinq sin-

gular solutions to Laplace's equation is a powerful

way of obtaininq solution•s to mary problems of poten-

tial theory-

The introduction of singularities often leads

to expressions for the induced field w-hich are in.-

versely proportional to R or R3 where R is the

distance between the singular point and the field

pomnt. In hydrodynamics, for instance the velocity

induced by a plane source is inversely proportional

to R while the use of the Biot-Savart Law leads to

expressions containing the inverse cube of R .JUsln

cylindrical coordinates (x,r,Q), we can write the dis-

tasce R as

R Wx-x) 2 r2 + r - 2rrcos(O0-0 5 (-)

where the bars (-) are used to denote the point of

the singularity Since a trigomonetrac Fourier Series

expansion in the angle 0 is used in many problems

2

adaptable to polar or cyitndrical coordinates, it is

of importance to study integrals of the type

'[ e-±ime f(cosus ij" d0 (2)

where the function f is analytic and p=l,3 Taking

into account that the integration interval is symmetric

with respect to 0=0 and corresponds to a full period

of the integrand, we see that we need, by substitution

of (-G)2"r , to examine essentially integrals of the

type

ctnp(a2 E f r/2 s2n, d- (3)"Iw/2 (a2+ 4sin2r)2/

2

for p=1,3 Here n is an integer and a is a real

constant,

An integral of similar type has been studied by

RiegelsI who related

3

K2 4 2 cos2n!Gn(. ) =_- (-i)•2"c- i -1- 2 n 3/2- . .. •d 4

for o-i-k2-1 -o a combination of complete elliptic

integrals of the first and second kind.. He also

gave a complete series representation of Gn(k2)

for small k the first terms of a series valid

for V near 1 , and tables of Gn(k ) for

o•-n-_7 . Gn(k 2 ) was encountered by Riegels 2 in

a study of Lbe flow past sernd~er, aimost axisvyr-

metric bodies, and by J. Weissinger3 n the aero-

dynamic theory of a ring airfoil.

The purpose of the present paper is to extend

Riegels' work by relating the integrals of Eqs0 t3)

and (4) to half order Legendre functions of the

second kind and to present complete series repre-

0 e--ations of these functions for values of the

argument in the neighborhood of unity and much

greater than unity. For convenient reference a

set of recursion formulas is included.

4

II. RELATION TO LEGENDRE FUNCTIONS

T"he integral +t, can ce transforrmed into a

double integral by using the Laplace transform of the

Bessel function of first kind and order zero, Jo., or

w/ 2'n2 = J cos >- d-- X

IV 2

4 e t 0T sin- ti dt (5)0

The order of integration can De interchanged essen-

tially because the right hand sade of Eq. (5) is

absolutely convergent it a/0 , We further observe

that J. can be expressed as an infinite sum of pro-

ducts of Bessel functions by Neumann s addition theoremt

So,

e at X

7112 2%_0c s2n1-. E mJM, (tlcos2m- ] d--, (6)-72Jpi-WI?-_o~

5

where cIT, is the Neumann factor:

M 2 m2 o1, m=o

Because the series is absolutely convergent, we car

integrate term by term. Since the set of cosine func-

tions is orthogonal in [-J/2,rI/2 1 , only the term for

mr 1n will give a contribution. We finally obtain

J - a 2> (t) dL (B)0 n

The Laplace transform that appears in Eq. (8i ts

expressible as a half order legendre function of the

second kind, Qn-½ The gereral formula from p. 389

of Ref. 4 is

e-Atj(Bt)J (Ct) dt0

(l . Q _ (A2,B2,c2)i2BC (9)Z

6

where it is assumed that the real parts of all of the

four numbers A+iB+iC are positive and that the real

part of v is greater than .

Applying Eq. (9) to ECt (8) with A= a and

B=C=l we find that,

r/2 cos2n'r -d 2(0j r2 ca , d-- ý Qn ._I(l+a2i2) (10)

•-.7/2 (a2+4sin2-T )1/2

Consequently differentiation with respect to the

2parameter a yields,

rr/2 cos2n-' d- Q (1 +a2/2) (11)

' 7/2 (a2t4sin 2-) 3/2 n - Q'/- /

The prime () denotes differentiation with respect

to the argument of Qn-½

*In the physical case of interest here, a is a realconstant. However, Eq. Lo) is valid for all a exceptalong the cut from -Ci tc *ic in the rcmplex plane.

7

By setting i(w/>t) and 2= 4/(a 2 _+4) in

Eq. (11) , we have upon comoarison with Eq. (4)

) =k- t--(4/ik,1)' t r Q ... (2//k 2 -1) (12)

Eq. (12) then relates the functior, Gn , tabulated

by Riegels, to the derivative of the half order

Legendre functions of the second kind.

8

III. SERIES REPRESENTATIONS OF Qn-X.(z)

First we consider the case of z near +I

Te Legendre functions of the second kind have loga-

rithmic singularities when the argument equals +1

Since in our application z=(l+a2 /2) and a is a

real constant, the singularity at z=+l is of spe-

cial interest.

A series representation of Qn.½(Z) valid near

z=l can be obtained by assuming a solution of the

form

c0

Qn-½(z) =X7- ans(z-1) +s=O

[In(z-l) ] Z bns(Z-1) s (13)5=0

to the corresponding Legendre differential equation,

0 = (l-z 2)Qwn-1½(z) - 2ZQ 'n½(z) +

(n 2 -¼) Qn-½ (z) (14)

9

Assume the complex z-piree to be cut from .1 to

-• along the real axis, if Eq- (13; is put into

Eq. (14), the following recurrence formulas Are

obtained by setting the coefficients in front of

each power of Mz-1) equal to zero:

n2-ns .. ns

2 (s+i.•2

b +2 (s~Il z

ns (

The first coefficients d., and bno can be

obtained from the foLlouing expression for Qn.½

nQ-

. , I (r,-A•'n P: '2iZ) +l ',n-9 7

O tz-ln (17)

10

Here II (n) =F(n+i) is Gauss0 function, Pn½ is a

Legendre function of the first kind, and 0 is the

Bachmann-Landau order of magaitude symbol. Using

the fact that Pn ½(l)=I , we can deduce from Eq. (17)

that

2 =- (18)•ns 2

a l1 n2 + ( U (o) - I' (n-u)ars 2 II (n-h)]

ns1

-2n2 - 2 7 (19)2 2j-1

Eqs. (13), (15), (16), (18) and (19) provide

the desired series expansion of Qn_12½(z) near z=+l

The corresponding expansion for Q'n_½(Z) is obtained

by differentiation of Eq. (13).

RiegelsI has calculated numerically the coeffi-

cients of a series expansion of Sn(k 2 ) for k2*1 .

By using Eq. (12), our results can be readily checked

with his.

11

Secondly, we consider the case for large z

In this case we can use a hypergeometric representa--

tion of Q n_(z) , for example,

Qn-ý½(z) F(n+½hJ, (z+1) L'h½- x

n! 2 n÷½

2 Flfn+½,n+½22n+l;2/(z+l) ) (20)

Or, after some rearrangement,

=n½z (n+s+i-½) ( 2/(z+l) }n+s+2 (21)

2 s=o sI(2n+s)f

for z>l Riegels' corresponding expansion for

Gn(k 2 ) can be obtained from Eq. (21).

12

IV. RECURSION FORMULAS FOR 0 (zi AND

Recursion formulas have been derived, for example,

in Ref. 6, and are given below for convenient refeiencet

Ln_½(z' = i__4½ Q'n+ (z) - Q'n.3/2(z) 1 (22)2n

Qn½z=(z) zQn-½(z) -Qn-3/2(z) 1 (23)z2-1

n+-1 (z) -= 2n_•z Qn-½ (z) - fl- Qn..3/2(z) (24)

Q? n+½ (z) = nz ,P (24)

2 n.-½ n- 1zl - -n-3/2(z) (25)

Qn--(z) = O~n..½(a) (26)

Q0n_½(Z) P Q'_n_½(z) (27)

The relationship between the half or,•uer Legendre

functions of the second kind and the complete ellip-

tic integrals o1 the first and second kind, or E(k)

and K1(k) respectively, is finally demonstrated by 6

13

2) MK(k) (28)

kt_½(lo a-_a2 E(k) (29)

a

where k=(4/(a 2 +4) )2 Hence, by using first the

recursion formula of Eq. (23) and then that of Eq.

(24) , it is possible to express the half order

Legendre functions in terms of the complete ellip-

tic integrals of the first and second kind for all

integer n

34

1.Rieqels, F., For-mein and Tabellen fur ein inder rauu'i±c'nen Poten tia-Lteor i e atit tretenoes

eljt~iches Integral Archiy der Matbhematik 2.1949-650, p. 117.

2. Riegels. F., Die Strom,-unq ur. -schlarhe,_fastdrehA~metr-ischeKLyjer~,ý !itteilunafen desMlax-Planck~-1nstituts fu~r Str6munqs forschung,Mr5, 195-2.

3. Weiss-Inger, J., Zur Aerodynarn-k CiesRinqfjýqelsinnopessibler Stror-una-i Z F Flugwiss 4

;aeft 314, 1956.

4. Watson, 'I'. N. , A Treatise on lthe "Theoiryof BesselFu-.:nct-ions, 'Cambridge, 195*3

5. Hobson, E. V. , Spherical1anQ:Ellipscsda1_[larrine.sCambridge University Press, 1931- p 206-.

6. Morse, P. M. and Feshbac-h- H-. Methods of Theoret--c'alltlyix~cs, McGraw-Hill, 1953 p 600,

AD

Reproduced

Armed Services Technical Information Aggii iARLINGTON HALL STATION; ARLINGTON 12 VIRGINIA

NOTICE: WHEN GOVERNMENT OR OTHER DRAWINGS, SPECIFICAT;DN' '

OTHER DATA ARE USED FOR ANY PURPOSE OTHER THAN IN CONN ut-hONWITH A DEFINITELY RELATED GOVERNMENT PROCUREMENT O1ILT -THE U. S. GOVERNMENT THEREBY INCURS NO RESPONSIBILITY, NOR.P pi:

OBLIGATION WHATSOEVER; AND THE FACT THAT THE GOVERNIMJJ IN' \THAVE FORMULATED, FURNISHED, OR IN ANY WAY SUPPLIED THE fLM DDRAWINGS, SPECIFICATIONS, OR OTHER DATA IS NOT TO BE REG1.iFE.- 31IMPLICATION OR OTHERWISE AS IN ANY MANNER LICENSING THE '10LDL.OR ANY OTHER PERSON OR CORPORATION$ OR CONVEYING ANY R1AlT . "iLPERMISSION TO MANUFACTURE, USE OR SELL ANY PATENTED INVE :TuO*:THAT MAY IN ANY WAY BE RELATED THERETO.

U NLSSF1