Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSgamma function. § It is straightforward to show that the hypergeometric seriesconverges in the range |x| < 1. It also converges at x =1ifc>a+ b andat x = −1 ifc>a+ b − 1. We also note that F(a, b, c; x) is symmetric in theparameters a and b, i.e.F(a, b, c; x) =F(b, a, c; x).The hypergeometric function y(x) = F(a, b, c; x) is clearly not the generalsolution to the hypergeometric equation (18.136), since we must also consider thesecond root of the indicial equation. Substituting σ =1− c into (18.138) anddemanding that the coefficient of x n vanishes, we find that we must haven(n +1− c)a n − [(n − c)(a + b + n − c)+ab]a n−1 =0,which, on comparing with (18.139) and replacing n by n + 1, yields the recurrencerelation(a − c +1+n)(b − c +1+n)a n+1 = a n .(n + 1)(2 − c + n)We see that this recurrence relation has the same form as (18.140) if one makesthe replacements a → a − c +1,b → b − c + 1 and c → 2 − c. Thus, provided c,a−b and c−a−b are all non-integers, the general solution to the hypergeometricequation, valid for |x| < 1, may be written asy(x) =AF(a, b, c; x)+Bx 1−c F(a − c +1,b− c +1, 2 − c; x),(18.143)where A and B are arbitrary constants to be fixed by the boundary conditions onthe solution. If the solution is to be regular at x =0,onerequiresB =0.18.10.1 Properties of hypergeometric functionsSince the hypergeometric equation is so general in nature, it is not feasible topresent a comprehensive account of the hypergeometric functions. Nevertheless,we outline here some of their most important properties.Special casesAs mentioned above, the general nature of the hypergeometric equation allows usto write a large number of elementary functions in terms of the hypergeometricfunctions F(a, b, c; x). Such identifications can be made from the series expansion(18.142) directly, or by transformation of the hypergeometric equation into a morefamiliar equation, the solutions to which are already known. Some particularexamples of well known special cases of the hypergeometric function are asfollows:§ We note that it is also common to denote the hypergeometric function by 2 F 1 (a, b, c; x). Thisslightly odd-looking notation is meant to signify that, in the coefficient of each power of x, thereare two parameters (a and b) in the numerator and one parameter (c) in the denominator.630