By F. W. J. Olver

A vintage reference, meant for graduate scholars mathematicians, physicists, and engineers, this ebook can be utilized either because the foundation for tutorial classes and as a reference software

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Z + 1) - (1/z) = ^(1 - z ) - π cotTTz = i ^ ( i z ) + \φ{\ζ+\) + In2. Ex. ). Ex. 3 From the preceding exercises derive ψϋ) = - y - 2 ln2. Ex. ). Deduce that when ζ is real and positive Γ(ζ) has a single minimum, which lies between I and 2. Ex. 5 If y is real, show that ου y_ 40 2 Introduction to Special Functions Ex. 6 Verify that each of the following expressions equals y: foo Γ1 Jo ' Jo ^ i// Too ^ / Ji e-' Jo \ i - e - ' Too/ i// /' ^-*\ , /; Deduce that y > 0. Ex. 1 The exponential integral is defined by 00 ^ - i — dt.

Prove that /(;c) = - l n j c - i y + ö(l) (^-^0), and exp(x^) d_ exp(;c2)/(;c) - n^'^ Jo'exp(w") ^« dx Hence establish that in terms of Dawson's integral and the exponential integral, fix) = n^i^F{x) - i cxpi-x') Ei{x^), t These results are due to Goodwin and Staton (1948) and Ritchie (1950), with a correction by Erdolyi (1950). 01) or its complement Γ(α,ζ), defined in the next subsection. Clearly y(a,z) is an analytic function of z, the only possible singularity being a branch point at the origin.

7 The final formula for the Gamma function in this section is an integral representation valid for unrestricted z. 01). The idea is due to Hankel (1864) and is applicable to many similar integrals. Consider I(z) = '(0 + ) e't-'dt, where the notation means that the path begins at / = — oo, encircles t = 0 once in a positive sense, and returns to its starting point; see Fig. 2. We suppose that the branch of t~' takes its principal value at the point (or points) where the contour crosses the positive real axis, and is continuous elsewhere.

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