# Evaluation of Some Integrals Following from $L_1$, the Constant of the Asymptotic Expansion of $\ln \Gamma_1 (x+1)$, Originating from Physics (QED)

## TL;DR

This paper compares three regularization methods in quantum electrodynamics to derive integrals related to the asymptotic expansion of a generalized gamma function, highlighting the mathematical and physical significance of the constant $L_1$.

## Contribution

It introduces a novel application of the constant $L_1$ from asymptotic gamma function expansion to evaluate integrals in physics and mathematics, using three regularization techniques.

## Key findings

- Derived new integrals involving $
abla ext{ln} \, ext{Gamma}_1(x+1)$
- Established connections between regularization methods and special functions
- Highlighted the role of the constant $L_1$ in integral evaluation

## Abstract

Comparison of three different regularization methods of calculating the one-loop effective Heisenberg-Euler Lagrangian of quantum electro-dynamics (QED) is employed to derive some interesting integrals involving the asymptotic expansion of $\ln \Gamma_1(x+1)$, the generalized $\Gamma$ function. Here it is the constant $L_1$ that will enable us to calculate some integrals which are useful in mathematics as well as in physics.