This paper is a pedagogical review of two mathematical classes of functions — completely monotone functions and Stieltjes functions — and how they appear in quantum field theory. Completely monotone (CM) functions obey an infinite hierarchy of sign constraints on all their derivatives. Stieltjes functions are a closely related, stronger class. The notes explain why these structures show up in several important QFT building blocks, including scalar Feynman integrals in the Euclidean region and certain amplitudes on the Coulomb branch of N=4 supersymmetric Yang–Mills theory.

The author surveys the mathematical definitions and basic properties. At a high level, a function is completely monotone if its derivatives alternate sign in a fixed way: the function is non‑negative, decreasing, and convex, and so on for higher derivatives. An equivalent characterization is that a CM function can be written as a Laplace transform of a positive measure — roughly, an integral of positive weight against decaying exponentials. Stieltjes functions satisfy even stronger analytic constraints in the complex plane. The notes collect closure properties and classical theorems (for example, variants of the Bohr–Mollerup and Carlson theorems) that explain uniqueness and analytic continuation under natural conditions.

The paper then discusses the physical and geometric origins of these positivity properties. In scattering amplitudes the constraints can follow from basic principles such as unitarity (probability conservation) and analyticity (well‑behaved dependence on complexified momenta). In perturbative Feynman integrals the same sign patterns can arise from the structure of parametric representations and Symanzik polynomials, which control the integrand geometry. The notes also review evidence that, in cases where an observable admits a representation as a “dual volume” or a Laplace‑type integral over a positive domain, complete monotonicity follows immediately.