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

Abstract

Exact formulas of generalized gamma functions, Γ 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 Γ 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.