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Integral Representations and Complete Monotonicity of Various Quotients of Bessel Functions

Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be expressed in terms of standard higher transcendental functions. Inverting a Laplace transform involves integrating a complex-valued function over a vertical line, and establishing the positivity of the resulting integral can be tricky. Sometimes asymptotic methods are helpful, see for example Fields and Ismail [6].