Generalized Legendre 1967

INTEGRALS INVOLVING GENERALIZED LEGENDRE FUNCTIONS by B. Meulenbeld* and H. S. V. de Shoo* 1. Introduction. in n The functions P~,n(z) and Qk ' (z), solutions of the differential equation: --" { m2 n 2 } (1-z 2) d2w - 2z dw + k(k+l) - - ~ w = 0, dz 2 dz 2(l-z) 2(l+z) introduced in [1] by KUIPERS-MEULENBELD, have been defined for all points of the z-plane in which a cross-cut exists along the real axis from 1 to -~, and in [2] for the real values of z on the cross-cut for -1 < z < 1. These functions will be called generalized Legendre functions, whereas P~(z) and Q~(z) which are the special cases for m=n, wile be denoted by associated Legendre functions. For the sake of brevity we put: = k+
89 fi = k-
89 3' = k+
89 5 = k-
89 In [3] is given a list of integral transforms whose kernels are associated Legendre functions. In this paper we want to extend this list to integral transforms with om,n Q~,n --k (x), (x), or combinations of these as kernels. In section 2 the result is given without proofs. The evaluation of most of the integrals is carried out by expressing P~,n(z) and Q~,n (z~ in terms of hypergeometric functions in conjunction with Lhe tabels in k3, 20.2]. A more detailed presentation may be found in [ 4]. In section 3 some in- verse transforms are given, making use of the inversion formulas of BRAAKSMA-MEULENBELD (see [5]). In the integrals, occurring in this section is integrated with respect to the lower parameter k. Finally, in section 4 we mention some useful relations, until now not yet published, between generalized and associated Legendre functions. 2. A. Laplace transforms with generalized Legendre functions. j e'Pt t c-I (t+l)-in p~,n(l+2t)dt = 0 For re=n: 2
89 P
89 r(~+i)r(-~) E(~+I, -7, c-
89 l-m: p) (1) Re c >
89 m, Re p > 0. f e-Pt re-1 -
89 m 1
89 (t+l) Pk (l+2t)dt = ~ ~- p sinrk E(k+l, -k, c-
89 (2) 0 Re c >
89 m, Re p > 0. For c=l-
89 in (2): J e-Pt [t(l+t)]-
89 i~ ~ -
89 pro-
89 e
89 (l+2t)dt 7r Kk+
89 (
89 0 Re m < 1, Re p > 0. * Technological University, Delft, the Netherlands. 286 B. Meulenbeld and H. S, V. de Snoo This transform has already been found in [3, 4.13(I)]. For m=0 in (2): ! -C e-Pt tc-1 Pk (l+2t)dt = - ~r p sin 7rk E(k+l, -k, e: 1: p) 0 Re e > 0, Re p > 0. J o e -ptt x+
89 - -~im,-,m.n (l+t)dt : F(a+ 1) ~ sin ~r E(-/~, 3'+1, k+m: re+I: 2p) + I"(6 + 1) 2m-n+l pX+m sinm~ sin azr 1 2"n+l pXsinmTr E(-a, 6+1, k: l-m: 2p) 1 j (3) Re k > 0, Re(k+m) > 0, For m=n (3) is equal to [3, 4.13(16)]. j 2t(t+y+z) e "pt t -
89 (t+y) -k-1 (t+z)k(t+y+z)
89 p ~,n (1 o (t+y) (t+z) = 2
89 e
89189 p-1 W k+
89189189189189189 n (pz) Re m < 1, yoz real > 0, For y=z=l and re=n=0 we get [3, 4.13(6)]. f o e "pt (i -e-t )
89 (l+e-t)~1:' k m' "n(et)dt : 2.P+
89 F(p-k)F(p+k+l) F(p+
89189189189 Re p>0. )d%= I Rep>0. (4) (5) p+k+l; p+-~m-~n+l, p+~m ~n+l;
89 3F2(p-k, p+
89 1 1 1 +l Re m >-1, Rep >-Re k-l, Re p > Re k. For m=n and ~=
89 we get [3, 4.13(11)]. pe t f -pt _l)
89 e (e t (--- l)~ = o p-2 F(p-~-k-
89 F(p-cr+k-
89 (6) = pp-
89 (p_2)-a F(p-~-
89189 3 F2 (p-a-k-
89 p+l, p-~+k+1-
89 p-~-
89 p-~+
89 ~-~) Re p > 0, Re m > -1, Re p > Re(
89 Re p > Re(
89 Integrals Involving Generalized Legendre Functions For m=n and ~=
89 we get [3, 4.13(12)]. f -pt -t in m,ntl -t
89 e (1-e) Pk t ~-2e )dr = 2 o F(p-
89189 F(-p-
89189 m,n (2e-r - 1)dr f e-pt (l_e-t) -
89 Pk o 287 (7) Re n >-1, Re p >
89 m. = 2
89 F(p+
89189 (8) l~(p-k-
89189 Re m < i, Re p >
89 r/[. B. Integral transforms with generalized Legendre functions. 1 m,n ~ xP(1-x)q(1-zx) -
89 Pk (1-2xz)dx = 0 = 2
89 z -
89 F(p-
89 F(i-m)F(p+q-
89 3F2 (/3+1, -T,~p-
89 -m,p+q-
89 z) Re p >
89 m-i, Re q > -1, z real:l z] < 1. (9) +i 7r 22
89 f _x 2)-
89 m.n(x)dx = (I Pk 1 -i cos ~n 7r Re m < I, ]Re n] < i. 1 I i 1" 1 I F(~-~a)F(~--UY)F(~fi+I)P(y6+I) (io) +I f (l_x)P(l+x)k_p_ 1 = 2 I~ r(k-p+
89189189 pk'n(x)dx F (fi+l)F(~+ 1 )P( -p -}m) Re(p-
89 > -i, Re(k-p)>
89 IRe n I. (11) § m,n (x)dx = f (1-x)P(I+x) qP k -1 F (q+
89189 = 2 p+q-i(m-n)+l r (1 -m)F (p+q-
89 3F2 (/3+1, --y, p-
89 l-m, p+q-
89 1). Re(p-
89 > -1, Re q+l >
89 ]Re n]. (12) § S -1 m,n (x)dx = ( 1 'x) -
89 ( 1 +x) q (u+x) "r P k q+
89 F(q+
89189 : 2 (u-1)'~ ' F(q-k-
89 )P(q+k-
89 3 F2 (q+
89 a, q.-
89 q-k-
89 q+k-
89 ]_2_~ )2 Re m < 1, Re q+l >
89 [Re n[, u not lying on the cut (-1,+1). (13) 288 B. Meulenbeld and H.S.V. de Shoo f (l-x) -
89 (l+x)qe -uxDm,n (x)dx = --k -1 = 2 q+
89 2F2 (q+
89 Rein< 1, (x-l) 4m 1 Rein< 1, P(q+
89189 9 e u ~ F (q-k-
89189 q-
89 q-k-
89 q+k-
89 -2u) Re q+l >
89 nl. (X+I)-P p m.n(x)dx = 21-m+
89 P(p+k+
89189 k F (p-
89 (p+
89 Re(p+k+}m) > 0, Re(p-k+
89 > 0. (14) (15) 1 (x-l) "
89 (x+l) "
89 (x+t)-2~-i p ~,n (x)dx = Rein< 1, F (a+2~+l)F(- 5+2~) -n-2~, -m-2~ (l+t) -Z-
89 ( 1 -t)-~-
89 Pk r(2~+l) IRe(2k+l) l < Re(m+n+4~+1), I t l < 1. (t) (16) ; e -~x (x-l) 1 Rein <1, -
89 In (x+l) "~ pk'n (x)dx Re a>0.
89 ... = 2
89 a vv
89189 (i7) e-aX(x_l)C-1 (x+l) m,n Pk (x)dx = 1 a
89 e-a - E(~+I, -% c-
89 l-m: 2a) F(~+I)F(-~) Re c >
89 m, Re a > 0. (18) (x-1)-
89 (x+l)-n eXt K [(l+x)t] P~n'n(x)dx =
89 1 2"
89 -
89 +1 1 1 1
89 (4t) = et~r cos
89 F(-k ~m-~)F(k+~m+~)t W
89189189 9 3~" Re m < 1, Re(-k+
89189 > 0, Re(k+
89189 > 0, l arg t I < ~--. (19) e-~im m.n Qk (x)dx = 1 r'(a+l)r(~/+l) F(1-
89 F(2k+2) r(k-
89 1 1 2 -~m+~n a F2 (/3+1, 5+1, k; 2k+2, k-
89 1) (20) Re k > 0, IRe m I < 2. Integrals Involving Generalized Legendre Functions 289 va f (x 2 -1) "
89 e 1 -2k-
89189 = 2 Re k > -1, -vim nm'n(X)d x = ~k 2 F(a+l)r(T+l) cos
89 i i i i 6+ f(~+ I)F(y~+I)F(~'+I)P(~ I) IRe ml <i. f m,. (x)dx : (X-1)P(X+I) "p'k'2 e'Vim Qk 1 -k-
89189 2 F (a+I)F(T+I)F (p-
89189 F(k+p-
89 (k+p+
89 Re k > -1, Re p >
89 m [ -1. (21) (22) m,n (x)dx = (X-1)P(x+l) q e -vim Q k 1 r(a+l)r(~+i) r(p-
89 F(2k+2") F(k-q-
89 2 p+q~m+
89 9 3 F2 (fl+l, 5+l,k-p-q; 2k+2, k-q-
89 1)
89 m I -1 < Re p < Re (-q+k). (23) pro.. Q~,n = k (x) (x)dx 1 ~in~ -m§ F(Z+
89 )r(~+
89 e 2 (~-k) (~+k+l) F(~-
89189 Re(~-k) > 0, Re(s > -1, Re m < 1. (24) One of the theorems of BRAAKSMA-MEULENfIELD [5] iS" be a real number with tel > ~Re m ~ ~lRe n l - 1, and ~(t) a Let kl function such that for all a > 1: ~o(t) (t-1)'i'
89 ml c L(1, a) if Re m / O, ~(t) (t-l) "
88 log(t-l) c La, a) if Re m = O, 99(t)t'l"kl e L(a, ~). Let further ~o(t) be a function of bounded variation in a neighbourhood of t=x(x > 1). Then ~o(t) satisfies the relations: kl+ ir 1 rj dkC2k+l) 'n (t)eVi QF" COdt : 2ri k I - i~ 1 290 and kl+ i=o f i ' -n 2-"~ j dk(Zk+l)e~imokm' (x) kl-i~ 1 B.Meulenbeld and H.S.V.de Shoo m,n (t)dt ~o(t)P k If we apply (25) to (24), we find after some simplifications: kt+ i~ 1 2 ~'--7 (-~ k -i~ = 2 m-n p~,n iX ) kl> -}Re me ~IRe n] ~+k+l 1, r(k+}(m+n)+1)r(k+
89 -m,-n (x)dk ) Pk F (k-}(m+n)+l)F (k-}(m-n)+l) (27) 2k 1 + 1 > 12Re .~ + 1 l, IRe ml <
88 > 1. Application of (26) yields: kt+i~ 1 1 1 f ( ~ m.n (x)dk Q~,n 2~i ~i ~-k $+k+l ') Qk = (x) (28) k I - ~ k I >
89 m +
89 n I - 1, 2Re ~ + 1 > I 2kl + 11, IRe ml< }, x > 1. This result can be found in a direct way by applying Cauchy's integral formula, since the asymptotic behaviour of (om,n "~k ix) as k--.=o on l arg k I <__ -~ with 0 < U < ~ is given by: Q~n,n (x) = km-
89 (x+\/x2-1)-ko(1) ( x > 1). Remark. The formula (27) may be considered as a continuous analogue of the c!assical Dougall series expansions. By applying (26) to (15) we obtain: kl+ i~ 1 ] (2k+l) (p+k+}m)r(p-k+}m-l)J im Q~m,-n 2~i k I - i= = 2"l+m'
89 F (p-
89 (p+}n) (x- 1) -
89 (x+ 1) "p k 1 > }Re m +
89 nl (x)dk = 1, Re(p+
89 -
89 > ]k 1+}1, Re m< ~, x > 1. (29) Application of (26) to (17) yields: kl+ im 1 15 m,n 2Vi (2k+!)W
89 (2a) e'~im Qk (x)dk = = 2
89 al+
89 -ax (x_l)
89 (x+l)
89 kl:~-}Re m + }IRe nl 1, Re a> 0, Rein>--34, x> 1. (30) For m=n (30) is transformed into [5, (8.27)]. Integrals Involving Generalized Legendre Functions 291 Some useful relation~ between the generalized and the associated Legendre functions are the following: -~m+
88 -
89 m P~'
89 (2z 2-1) = 2 z P 2k+
89 (Z). (31) (the upper or lower sign according as Imz ~ 0). 3 1 Q~n,
89 (2z 2_1) = 2 z-
89 2k.,
89 (z). 3 3 "
89 9 z2+ 1 2"~m+4e 2~ik z (z 2 1)
89 ~kDm'
89 (Z--~_ 1 ) = yr~F(_m_2k_
89 (33) -2k-1 Q-
89 (z). (34) Formulas (31) and (33) hold for Re z ~ 0, z not on the cross-cut, and for -1 < z < 1, z r 0, Formula (32) holds for Re z ~,0, z not o~. the cross-cut, and for -1 < z < 1, z ~ 0 after omitting the factor e ~lm Formula (34) holds for z not on the cross-cut. References. 1. L.KUIPERS and B.MEULENBELD 2. B. lvigULENBELD 3. A.ERDELYI a.o. 4. H.S.V. DE SNOO 5. B.L.J.BRAAKSMA and B. MEULENBELD On a generalization of Legendre's associated 9 differential equation I and II, Proc. Kon.Ned.Ak, v.Wet., Amsterdam 60, 436-450 (1957). Generalized Legendre's associated functions for real values of the ar- gument numerically less than unity, Proc. Kon. Ned. Ak. v. Wet., Am- sterdam 61, 557-563 (1958). Tables of integral transforms, vol. 1, ,New York, 1954. Integral transforms and series involving generalized Legendre functions, Report Technological University Delft, 1967. Integral transforms with generalized Legendre functions as kernels, Compositio Mathematica (will appear in 1967). [Received May 26, 1967]