Gaussian integrals and their applications

Oscar A. Nieves is a theoretical physicist working in applied superconductivity research. His research focuses on the modelling of superconducting electronics, quantum sensors and photonic devices using semi-classical methods. Oscar completed his PhD in Theoretical Physics from the University of Technology Sydney (UTS) in Australia, with the thesis entitled “Noise Dynamics in Stimulated Brillouin Scattering”. Here, he investigated the role of thermal noise and laser noise in the interaction between light pulses and acoustic waves in dielectric waveguides, otherwise known as the stimulated Brillouin scattering interaction (SBS); and the possible limitations this has for the realization of photonic devices that exploit this interaction, including: amplifiers, filters and short-term memory storage devices. During this degree, Oscar studied stochastic processes very closely, and frequently came across things like the normal and log-normal probability distributions, from which his interest in Gaussian integrals emerged. Prior to his PhD, he conducted research in simulating the optical response of one-dimensional periodic metasurfaces, investigating the relation between surface curvature and reflectance properties using the Finite-Difference Frequency-Domain (FDFD) method, which he programmed in MATLAB. His main areas of interest include: numerical methods for physics applications, numerical solution of partial differential equations, stochastic modelling, quantum devices, electromagnetic theory and machine learning. He also enjoys writing Python codes for solving different types of problems, especially in relation to stochastic dynamics. Aside from his day-to-day research activities, Oscar also finds happiness in teaching applied mathematics and physics. He has taught a variety of university courses ranging from introductory Calculus and statistics to partial differential equations and complex analysis.

1. Elementary Results. 1.1. The Gaussian Function. 1.2 The Gaussian Function in Multiple Dimensions. 1.3. An Unexpected Derivation. 1.4. Powers of Incomplete Squares. 1.5. The Fourier Transform of a Gaussian. 1.6. The Error Function. 2. Compound Gaussian Integrals. 2.1. Polynomials of Even Order. 2.2. Polynomials of Odd Order. 2.3. Polynomials of Non-Integer Order and the Gamma Function. 2.4. Higher Order Gaussian Integrals. 3. Multivariate Gaussian Integrals. 3.1. Exponents of Scalar Terms. 3.2. Exponents of Quadratic Forms. 4. Applications. 4.1. Fresnel Integrals. 4.2. The Normal Distribution. 4.3. The Log-Normal Distribution. 4.4. Brownian Motion and Diffusion. 4.5. Path Integrals in Quantum Theory.