AN EXPANSION FOR LAPLACIAN INTEGRALS 



565 



For the computation of P{Ha, T') recourse may be had to the 

 extensive "Incomplete Gamma Function" Tables edited by Karl 

 Pearson which give the ratio of the incomplete to the complete function. 

 When Ht is a positive integer we have 



the well known Poisson Exponential Binomial Limit for which short 

 tables will be found in Pearson's "Tables For Statisticians And 

 Biometricians " and in T. C. Fry's "Probability and Its Engineering 

 Uses." 



Applications 

 I 

 Consider the incomplete Beta Function 



Kp) = r (1 - xy-^x^-^dx 



Jo 



for positive integral values of n and c such that n is large compared 

 with c. Its modified Laplacian expansion is 



I{p) = {c - \)\{n + w)-'=Z(w + w)-^Aim, c - l)P(c + m, T2), 

 where T2 = — (n + w) log (1 — p) and, setting w = {d — c + l)/2,i 



1. 

 c\d 

 1/2' 



c+ 1 

 2 



c + 2 

 \ 3 

 c + 3 

 4 



-(:r)[ 



(c - 1) + 3d^ 

 12 



(c - l)d 4- d' 



5(c - ly - (2 - 30d^)(c - 1) + 15^^ 

 240 



As many more coefficients as one desires can be obtained by means 

 of the equations 



c — I -\- m 



fc=o \ m + k 



Q{m + \,k) = {m-\- k)Q(m, k - l) + (w + k)Q{m, k), 

 Q(o, 0) = 1. 

 ^ See Appendix II for the details of the analyses. 



