Appendix. the Mellin Transform and Related Analytic Techniques

1. The generalized Mellin transformation The Mellin transformation is a basic tool for analyzing the behavior of many important functions in mathematics and mathematical physics, such as the zeta functions occurring in number theory and in connection with various spectral problems. We describe it first in its simplest form and then explain how this basic definition can be extended to a much wider class of functions, important for many applications. Let ϕ(t) be a function on the positive real axis t > 0 which is reasonably smooth (actually, continuous or even piecewise continuous would be enough) and decays rapidly at both 0 and ∞, i.e., the function t A ϕ(t) is bounded on R + for any A ∈ R. Then the integral ϕ(s) = ∞ 0 ϕ(t) t s−1 dt (1) converges for any complex value of s and defines a holomorphic function of s called the Mellin transform of ϕ(s). The following small table, in which α denotes a complex number and λ a positive real number, shows how ϕ(s) changes when ϕ(t) is modified in various simple ways: ϕ(λt) t α ϕ(t) ϕ(t λ) ϕ(t −1) ϕ ′ (t) λ −s ϕ(s) ϕ(s + α) λ −s ϕ(λ −1 s) ϕ(−s) (1 − s) ϕ(s − 1). (2) We also mention, although we will not use it in the sequel, that the function ϕ(t) can be recovered from its Mellin transform by the inverse Mellin transformation formula ϕ(t) = 1 2πi C+i∞ C−i∞ ϕ(s) t −s ds , where C is any real number. (That this is independent of C follows from Cauchy's formula.) However, most functions which we encounter in practise are not very small at both zero and infinity. If we assume that ϕ(t) is of rapid decay at infinity but grows like t −A for some real number A as t → 0, then the integral (1) converges and defines a holomorphic function only in the right half-plane ℜ(s) > A. Similarly, if ϕ(t) is of rapid decay at zero but grows like t −B at infinity for some real number B, then ϕ(s) makes sense and is holomorphic only in the left half-plane ℜ(s) < B, while if ϕ(t) has polynomial growth at both ends, say like t −A at 0 and like t −B at ∞ with A < B, then ϕ(s) is holomorphic only in the strip A < ℜ(s) < …