AN EXPANSION FOR LAP LA CI AN INTEGRALS 



569 



But df{x)ldt = f{x){dxldt). Therefore, taking /(x) such that /'(a:) 

 = <i>w{x), 





dt 



A . = [D/g-^'+^^<f>Mlx=x, D.' = — , 



W 





or 



(8) 







[5 - (m + l)z;]! 



Beta Function 

 For the incomplete Beta Function we have, 



X=0, u^O, F = 1, g(0) = 1, 



c — 1 + r 



</>(:x;) = x-Kl -x)-=i:(' J"^") 



Therefore, 



S—(.C—1) „.,v 





-^= E ^E 



(5 - z;) ! 



or 





[5 - (c - 1) - z; - ry. 



Since -44=0 for 5 < c — 1, set s — c — 1 -\- m; then interchanging 

 y and m — v we obtain 



^^^ (c-l+m)! i>(^^ -t^)!r=o V c-1 ; {v-r)l 



To evaluate (9) we may have recourse to the formula * 



(- i)^(^)z)/^g^ 



(10) 

 taking 



..^.-s,-l,-)i 



(5 + X)g^+^ 



M = V — r, 

 S = c -\- V. 



* See formula 25, page 15, of Schlomilchs Hoheren Analysis, Band II, Dritte 

 Auflage, 1879. I am indebted to Mr. J. Riordan of the Department of Development 

 and Research, American Telephone and Telegraph Company, for calling my attention 

 to this formula. 



