Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSsecond solution to (18.147) as a linear combination of (18.148) and (18.149) givenbyU(a, c; x) ≡π []M(a, c; x) M(a − c +1, 2 − c; x)− x1−c .sin πc Γ(a − c +1)Γ(c) Γ(a)Γ(2 − c)This has a well behaved limit as c approaches an integer.18.11.1 Properties of confluent hypergeometric functionsThe properties of confluent hypergeometric functions can be derived from thoseof ordinary hypergeometric functions by letting x → x/b and taking the limitb →∞, in the same way as both the equation and its solution were derived. Ageneral procedure of this sort is called a confluence process.Special casesThe general nature of the confluent hypergeometric equation allows one to writea large number of elementary functions in terms of the confluent hypergeometricfunctions M(a, c; x). Once again, such identifications can be made from the seriesexpansion (18.148) directly, or by transformation of the confluent hypergeometricequation into a more familiar equation for which the solutions are alreadyknown. Some particular examples of well known special cases of the confluenthypergeometric function are as follows:M(a, a; x) =e x ,M(1, 2; 2x) = ex sinh x,xM(−n, 1; x) =L n (x), M(−n, m +1;x) = n!m!(n + m)! Lm n (x),M(−n, 1 2 ; x2 )= (−1)n n!(2n)! H 2n(x), M(−n, 3 2 ; x2 ), = (−1)n n!2(2n +1)!M(ν + 1 2 , 2ν +1;2ix) =ν!eix ( x 2 )−ν J ν (x), M( 1 2 , 3 2 ; −x2 )=√ π2x erf(x),H 2n+1 (x),xwhere n and m are integers, L m n (x) is an associated Legendre polynomial, H n (x) isa Hermite polynomial, J ν (x) is a Bessel function and erf(x) is the error functiondiscussed in section 18.12.4.Integral representationUsing the integral representation (18.144) of the ordinary hypergeometric function,exchanging a and b and carrying out the process of confluence givesM(a, c, x) =Γ(c)Γ(a)Γ(c − a)634∫ 10e tx t a−1 (1 − t) c−a−1 dt,(18.150)