Exact and useful formulas of generalized gamma functions occurring in finite diffraction theory

摘要

Exact formulas of generalized gamma functions, Gamma(m)(u, z), occurring in finite diffraction theory are derived in closed form for arbitrary m, u = n + 1/2 (m and n are non-negative integers), and for both real and complex arguments z. For m = 1 and real argument z, the formula consists of polynomials and the complementary error function. And, for m = 1 and purely imaginary argument z occurring in the Wiener-Hopf integral equation for a finite diffraction problem, the formula is expressed by polynomials and the Fresnel integral which is a well-known function in mathematical theory of diffraction. The formulas for an arbitrary positive integer m are also obtained simply by differentiating Gamma(m)(u, z) with respect to z. These exact formulas are graphically shown and compared with Kobayashi's asymptotic formulas for various m and n values.