Calculation of the Gaussian Integral by Real and Complex Approaches

The Gaussian integral is a very important integral in real analysis. It has many applications in various areas, including probability theory, statistical mechanics, and engineering. Nevertheless, the calculation of Gaussian integral is not easy and some tricks should be involved. To this end, this paper mainly focused on the Gaussian integral. There are three main parts about testifying of the Gaussian integrals, background and application. The first part is focused on the background and the meaning of Gaussian integral. The second part is mainly focused on three methods to testify Gaussian integral. Pertaining these methods, the first is about the real method and double integral, the second is about complex variables, and the third is about generalized integral. The last one is about to testify this integral by using normal integral under one variable integral. The third part of the paper is about the application of this integral. These include the extend of Gaussian integral, the general form of Gaussian which is more complex, and the Gaussian integral in thermodynamics, aiming to get general formulas on heat conduction.