Special Functions from a Complex Viewpoint

TL;DR

The notes develop a cohesive complex-analytic framework for classical and higher-order gamma-type functions, leveraging Nevanlinna–Pick and Stieltjes classes to obtain integral representations, positivity properties, and monotonicity. By studying inverses, Barnes' G-function, and multiple gamma functions, the work reveals deep connections between completely monotone, Bernstein, and generalized Stieltjes functions, including their higher-order variants and remainder analyses in asymptotic expansions. The approach yields operator-theoretic and probabilistic interpretations (e.g., kernels, convolution semigroups) and provides practical tools for analyzing gamma-function ratios through structured function classes. Overall, the course demonstrates how complex-analytic methods give precise positivity and monotonicity results for a broad family of special functions with applications to inequalities and functional calculus.

Abstract

This document contains the lecture notes for a mini-course on special functions from a complex viewpoint given at the OPSFA Summer School OPSFA-S10 2024, in the period July 29th -- August 2nd, 2024. The summer school was hosted by Section of Mathematics -- Luciano Modica at the Uninettuno University.

Figures (9)

• Figure 1: Graph of $\Gamma$ on $(0,\infty)$
• Figure 2: The function on the positive line; its imaginary part in the upper half-plane; the boundary behaviour on the negative axis; the derivatives on the positive axis.
• Figure 3: The graph of $d$
• Figure 4: $\Gamma$ on the real line
• Figure 5: The domain $\log \Gamma (\mathbb H)$

Theorems & Definitions (95)

• Example 1.1
• Proposition 1.3: Local version of the ordinary maximum principle
• Proposition 1.4: A maximum principle in a bounded domain
• Definition 1.5
• Example 1.6
• Proposition 1.7: Maximum principle for subharmonic functions
• proof
• Proposition 1.8: Maximum principle in unbounded domains
• proof
• Proposition 1.9: Growth times opening theorem