Mathematical Methods for Physics and Engineering - Matematica.NET

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONSwhich converges provided c>a>0.◮Prove the result (18.150).Since F(a, b, c; x) is unchanged by swapping a and b, we may write its integral representation(18.144) asΓ(c)F(a, b, c; x) =t a−1 (1 − t) c−a−1 (1 − tx) −b dt.Γ(a)Γ(c − a) 0Setting x = z/b and taking the limit b →∞,weobtainM(a, c; z) =Γ(c)Γ(a)Γ(c − a)∫ 10∫ 1t a−1 (1 − t) c−a−1 limb→∞(1 − tz bSince the limit is equal to e tz , we obtain result (18.150). ◭) −bdt.Relationships between confluent hypergeometric functionsA large number of relationships exist between confluent hypergeometric functionswith different arguments. These are straightforwardly derived using the integralrepresentation (18.150) or the series form (18.148). Here, we simply note twouseful examples, which readM(a, c; x) =e x M(c − a, c; −x), (18.151)M ′ (a, c; x) = a M(a +1,c+1;x), (18.152)cwhere the prime in the second relation denotes d/dx. The first result followsstraightforwardly from the integral representation, and the second result may beproved from the series expansion (see exercise 18.19).In an analogous manner to that used for the ordinary hypergeometric functions,one may also derive relationships between M(a, c; x) and any two of thefour ‘contiguous functions’ M(a ± 1,c; x) andM(a, c ± 1; x). These serve as therecurrence relations for the confluent hypergeometric functions. An example ofsuch a relationship is(c − a)M(a − 1,c; x)+(2a − c + x)M(a, c; x) − aM(a +1,c; x) =0.18.12 The gamma function and related functionsMany times in this chapter, and often throughout the rest of the book, we havemade mention of the gamma function and related functions such as the beta anderror functions. Although not derived as the solutions of important second-orderODEs, these convenient functions appear in a number of contexts, and so herewe gather together some of their properties. This final section should be regardedmerely as a reference containing some useful relations obeyed by these functions;a minimum of formal proofs is given.635