By Krebs A., Stephan E.P.

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If 9t(c — a — b) > 0, the value at z = 1 is given by Special cases are The power series for the Gauss hypergeometric function provides a simple and efficient means for the computation of this function, when z is properly inside the unit disk. 2 Chapter 2. 73) converges inside the unit disk, and for numerical computations we can use only the disk \z\ < p < 1, with p depending on numerical requirements, such as precision and efficiency. Other power series are available, however, to extend this domain.

3. Watson's lemma: sectors in the t-plane and the z-plane. 14 (incomplete gamma function T(a, z)). tion For the incomplete gamma func- we expand For this function a = n, and we obtain We remark that this asymptotic expansion holds uniformly in the sector |ph z\ < \it — S. This range is much larger than the usual domain of definition for the incomplete gamma function, which reads |phz | < n. 13. 113) is usually written in the form 38 Chapter 2. Convergent and Divergent Series Observe that the exponential integral is a special case.

When we expand the solutions at a particular point, say, the origin, in many important cases the theory of regular and singular points is needed. 3. 15) governs the well-known functions of mathematical physics. 2. Differential equations and Frobenius series solutions 19 • the Bessel differential equation • the Legendre differential equation • the confluent hypergeometric (Kummer) differential equation • the Whittaker differential equation • the Hermite differential equation • the Weber or parabolic differential equation We will meet these equations in the later sections.