Dirichlet integral, residue theorem, Feynman’s trick, gamma function
Abstract
Dirichlet integral, as the integration of sine function over x from positive infinity to 0, or negative infinity, is known as one of the most important concepts in mathematical analysis. In probability and statistics, Dirichlet integral is used in calculating expectation and variance. In physics, Dirichlet integral is used in describing the movement of a charged particle in specific electromagnetic fields, and the wave function of particles in quantum physics. In computer science, Dirichlet integral is used in the optimization of neural network. In economics and management, Dirichlet integral is used in the optimization of production and distribution, the estimation of the risk and return in investment. In this paper, Dirichlet Integral and its generalization are studied. This paper provides several methods including residue theorem and Feynman’s trick to determine the value of Dirichlet integral. Meanwhile, this paper extends Dirichlet integral to the case of n -th power and deduces a general solution by gamma function for the integral.
