Reproduced ARLINGTON HALL STATION; ARLINGTON 12 VIRGINIA

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1 I a ' Reproduced armed Services Technical Information figency ARLINGTON HALL STATION; ARLINGTON 12 VIRGINIA NOTICE: WHEN GOVERNMENT OR OTHER DRAWINGS, SPECIFICATIONS OR OTHER DATA ARE USED FOR ANY PURPOSE OTHER THAN IN CONNECTION WITH A DEFINITELY RELATED GOVERNMENT PROCUREMENT OPERATION, THE U. S. GOVERNMENT THEREBY INCURS NO RESPONSIBILITY. NOR ANY OBLIGATION WHATSOEVER; AND THE FACT THAT THE GOVERNMENT MAY HAVE FORMULATED, FURNISHED, OR IN ANY WAY SUPPLIED THE SAID DRAWINGS, SPECIFICATIONS, OR OTHER DATA IS NOT TO BE REGARDED BY IMPLICATION OR OTHERWISE AS IN ANY MANNER LICENSING THE HOLDER OR ANY OTHER PERSON OR CORPORATION, OR CONVEYING ANY RIGHTS OR PERMISSION TO MANUFACTURE, USE OR SELL ANY PATENTED INVENTION THAT MAY IN ANY WAY BE RELATED THERETO. 5N'SSFE

2 ii 'II L _-_ APOP- THERM ADVANCED RESEARCH H A ra E w Y 0R K

3 EXPRESSION AS A LEGENDRE FUNCTION, AN ELLIPTIC INTEGRAL OCCURRING IN WING THEORY OF by B. 0. U. Sonnerup TAR-TN 59-! November 1959 Submitted to the Air Branch, Office of Naval Research in Partial Fulfillment of Contract Nonr-2859(00) Donald Earl Ordway Head, Aerophysics Section Approved: OK A. Ritter Director Therm Advanced Research

4 Reproduction in whole or in part is uermitted for any purpose of the United States Government

5 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 argument 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 notedii

6 SYMBOL:Z A,BC a constants 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 m n modulus of elliptic integrals real integer nutiber real integer number Pv(Z) Legendre function of first kind Qv(z) R (x,r,&) Legendre function of second kind spatial distance between singularity and field point cylindrical coordinates U: (n) Gauss' function II n) = ['(n+l) v complex number iii

7 I, I NTRODUCTI ON The method of supcrrmpos-_riq or dastributinq singular solutions to Laplace's equation is a powerful way of obtaininq solution s to mary problems of potential theory- 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 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 distasce R as R Wx-x) 2 r2 + r - 2rrcos(O0-0 5 (-) where the bars (-) the singularity are used to denote the point of Since a trigomonetrac Fourier Series expansion in the angle 0 is used in many problems

8 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

9 3 K2 Gn(. =_- (-i) 2"c- ) 4 2. cos2n! 2 i.. n -1- d 3/2-4 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 axisvyrmetric bodies, and by J. Weissinger3 n the aerodynamic theory of a ring airfoil. The purpose of the present paper is to extend Riegels' work by relating the integrals of Eqs 0 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.

10 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 'n2 w/ 2 = J cos >- d-- X IV 2 4 e t T 0 sin- ti dt (5) 0 The order of integration can De interchanged essentially 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 products of Bessel functions by Neumann s addition theoremt So, e at X 7112 %_0c s2n1-. E mjm, 2 (tlcos2m- ] d--, (6) -72Jpi -WI?-_o~

11 5 where cit, is the Neumann factor: M 2 m2 o 1, m=o Because the series is absolutely convergent, we car integrate term by term. Since the set of cosine functions is orthogonal in [-J/2,rI/2 1, only the term for mr 1 n 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 0 e-atj(bt)j (Ct) dt (l. Q _ (A2,B2,c2)i2BC (9) Z

12 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(0 j r2 ca, d-- ý Qn._I(l+a2i2) (10) -.7/2 (a2+4sin2-t )1/2 Consequently differentiation with respect to the 2 parameter 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 real constant. However, Eq. Lo) is valid for all a except along the cut from -Ci tc *ic in the rcmplex plane.

13 7 By setting i(w/>t) and 2= 4/(a 2 _+4) in Eq. (11), we have upon comoarison with Eq. (4) ) =k---(4/ik,1)' t t Q r... (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.

14 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 logarithmic singularities when the argument equals +1 Since in our application z=(l+a 2 /2) and a is a real constant, the singularity at z=+l is of special 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)

15 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: -ns.. ns n2 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)

16 10 Here II (n) =F(n+i) is Gauss 0 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 ns =- 2 (18) a ars l2 1 n2 + ( U (o) - II I' (n-u) (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 coefficients of a series expansion of Sn(k 2 ) for k 2 *1. By using Eq. (12), our results can be readily checked with his.

17 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).

18 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 = i 4½ Ln_½(z' 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) Q'_n_½(z) P (27) The relationship between the half or, uer Legendre functions of the second kind and the complete elliptic integrals o1 the first and second kind, or E(k) and K1(k) respectively, is finally demonstrated by 6

19 13 2) MK(k) (28) a-_a2 kt_½(lo 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 elliptic integrals of the first and second kind for all integer n

20 34 1.Rieqels, F., For-mein and Tabellen fur ein in der rauu'i±c'nen Poten tia-lteor i e atit tretenoes eljt~iches Integral Archiy der Matbhematik , p Riegels. F., Die Strom,-unq ur. -schlarhe,_fast dreha~metr-ischeklyjer~,ý!itteilunafen des Mlax-Planck~-1nstituts fu~r Str6munqs forschung, Mr5, Weiss-Inger, J., Zur Aerodynarn-k CiesRinqfjýqels innopessibler Stror-una-i Z F Flugwiss 4 ;aeft 314, Watson, 'I'. N., A Treatise on lthe "Theoiryof Bessel Fu-.:nct-ions, 'Cambridge, 195*3 5. Hobson, E. V., Spherical1anQ:Ellipscsda1_[larrine.s Cambridge University Press, p Morse, P. M. and Feshbac-h- H-. Methods of Theoret--c'all tlyix~cs, McGraw-Hill, 1953 p 600,

21 AD Reproduced Armed Services Technical Information Aggii ARLINGTON HALL STATION; ARLINGTON 12 VIRGINIA i NOTICE: WHEN GOVERNMENT OR OTHER DRAWINGS, SPECIFICAT;DN' ' OTHER DATA ARE USED FOR ANY PURPOSE OTHER THAN IN CONN ut-hon WITH 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' \T HAVE FORMULATED, FURNISHED, OR IN ANY WAY SUPPLIED THE flm D DRAWINGS, SPECIFICATIONS, OR OTHER DATA IS NOT TO BE REG1.iFE.- 31 IMPLICATION OR OTHERWISE AS IN ANY MANNER LICENSING THE '10LDL. OR ANY OTHER PERSON OR CORPORATION$ OR CONVEYING ANY R1AlT. "il PERMISSION TO MANUFACTURE, USE OR SELL ANY PATENTED INVE :TuO*: THAT MAY IN ANY WAY BE RELATED THERETO. U NLSSF1