Nearly 200 problems, each with a detailed, worked-out solution, deal with the properties and applications of the gamma and beta functions, Legendre polynomials, and Bessel functions. The first two chapters examine gamma and beta functions, including applications to certain geometrical and physical problems such as heat-flow in a straight wire. The following two chapters treat Legendre polynomials, addressing applications to specific series expansions, steady-state heat-flow temperature distribution, gravitational potential of a circular lamina, and application of Gauss's mechanical quadrature formula with pertinent table. The final chapters explore Bessel functions, discussing differentiation formulas, generating functions, relations to Legendre polynomials, and other applications.

This volume constitutes a useful tool for professional engineers and experimental physicists. Students of mathematics, physics, and engineering will particularly benefit from the book's expanded solutions.

Problems: Differentiation Formulas V-1. Show that When p is such that xp is not real for x negative, we shall exclude negative values of x from consideration. When p is such that negative powers of x are contained in any of the series expansions involved, then x = 0 will naturally be excluded. Case 1. p is not a negative integer. Then by Eq. (V-0.2) we have Also by Eq. (V-0.2) we get Termwise differentiation is valid in obtaining Eq. (V-1.3) since the resulting series converges

powers of h we naturally try making use of Maclaurin’s series for eu, namely Then, by Eq. (V-22.1), we may write Since both series in Eq. (V-22.2) converge by Eq. (V-22.1) for all x together with all h ≠ 0 and, for each choice of x, converge uniformly on any closed finite interval of values of h not containing h = 0, we may multiply termwise the two series and collect together terms in like powers of h; and the resulting series so formed will converge to G(h, x) for all x together with all

where αn, n = 1, 2, 3, · · · are the positive zeros of J0(x). Assuming the series expansion in Eq. (V-26.1) is termwise integrable over B when each term thereof is multiplied by an arbitrary J0(αkx), show that the coefficients An, n = 1, 2, 3, · · · are given by This problem is analogous to Prob. III-18. Multiplying both sides of Eq. (V-26.1) by xJ0(αkx) and integrating over B, we get Every integral on the right in Eq. (V-26.3) vanishes by the orthogonality property shown in Prob. V-24

constant to be determined by boundary conditions, (b) is a function of one variable only, namely distance from longitudinal axis of reactor, that is, is symmetric with respect to this axis, (c) if cylindrical coordinates (r, θ, z) be taken with the z-axis in the longitudinal axis of the reactor, then the flux satisfies the conditions In terms of cylindrical coordinates Eq. (VI-18.1) is But, the given condition (b) means that is independent of both θ and z. This makes Eq. (VI-18.4)

an iterated integral, thence to the equivalent double integral in rectangular coordinates, thence to polar coordinates, and finally to a product of integrals. Let us ask ourselves, “If that scheme was successful in evaluating the product of , might we not expect it to yield something useful not only for a particular such product but also for the general product Γ(x)Γ(y)?” At any rate it is worth trying to see what happens. We start with the product Γ(p)Γ(q) instead of Γ(x)Γ(y), because we shall