184 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



8.540 The general solution of equation 8.510 may be written: 



4> = F(p, q, & = C M + C * N , 



M 



N. 



= I W-Hi - u)- l 



Jo 







T(s) \ 



s = P + q, 



T(p)T(q) | _P { ,X* + i) P P(P+* 

 Ts * " si). s(s+i) 2! s(s+i) 



. (p - i)(/> - 



' 



(p - i)(p - 2) (p-n- i)(g)(q + i) (q + n - 2) 



(n - i) If"- 1 



p(p - i)(p - 2) ....(/>- n}q(q + i)(q + 2) . . . . (q + n - i) ] 

 where o < p < i and the real part of is positive. 



THE COMPLETE SOLUTION OF EQUATION 8.510 IN SPECIAL CASES 



8.550 p>o, q>o, real part of >o: 



F(P, q, )-*!/ MP-^I - u^^e-^du + c 2 e- / (i + u)P- l u- l e-t u d 



Jo Jo 



8.551 ^>o, g>o, J<o: 



^(A ^ I) = ^i / w^K 1 - u) q - 1 e~t u du+ c 2 I u p ~ l (i + u) q - l < u du. 



8.552 ^<o, ?<o, ?>o: 



{/*! /*00 



d I (i-u)-Pu-ve-t u du + c 2 e-t I 

 Jo Jo 



8.553 ^<o, g<o, ^<o: 



{/*r /*oo 



ci I (i - uJ-Pu-te-^du + c 2 I 



8.554 



p - m + r, where m is a positive integer and r a proper fraction. 



d m I 1 



