The Functions of Mathematical Physics
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Comprehensive textbook provides both mathematicians and applied scientists with a detailed treatment of orthogonal polynomials, principal properties of the gamma function, hypergeometric functions, Legendre functions, confluent hypergeometric functions, and Hill's equation. Lucid and useful presentations for anyone working in pure or applied mathematics or physics.
evaluate the above for a particular value of z, say z = 1/m. Then To evaluate the latter we can write by use of III, 2.1. If we let so that and note that we have Since the εk are certain roots of unity we can write so that This result combined with (7) again yields the multiplication theorem. 4. Beta Functions A function closely related to the gamma function is the beta function, defined by THEOREM. Proof. We now introduce the new coordinates so that
inside the orthogonal circle and invariant under the group of all even inversions generated by the three possible inversions in each of the three sides of the triangle. Such a function is known as an automorphic function. In our preceding discussion we took it for granted that n, m and l are positive, but finite integers. Certain interesting special cases arise if one of them is allowed to become infinite. In this case one vertex becomes a cusp. These cases are intimately related to the theory
evaluate the two remaining integrals we select that branch of our integrand for which The second integral then becomes Finally by 2.6, thus establishing (7). 8. Asymptotic Expansions for Large Argument To obtain suitable asymptotic expansions for Hν(1)(z), for example, we return to 7.6 and let t = 1 + iw. Then where C is the following contour. The integrand in (1) has two branch points, at w = 0 and w = 2i, respectively. For Re z > 0 and Re the integral in (1) converges. We
equation and E must be so selected that u is bounded for all x. We know that this can happen if and only if Δ(E), the discriminant is bounded between + 2 and − 2. The permissible energy states of (1) must therefore lie on the stability regions of equation (3). According to Floquet's theorem (3) will have solutions of the form Where p(x) is periodic and λ depends on E. To deduce the relationship between λ and E we see that, if R is the period of V(x), then so that For λ to be real in (4),
Watson's lemma, 70 Weierstrass approximation theorem, 11 ff Whittaker's equation, 191 Zeros, of Bessel functions, 249 of Jacobi polynomials, 45 ff of orthogonal polynomials, 14 ff, 23 ff