Articles containing proofs | Gaussian function | Integrals | Theorems in analysis
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related to both the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in statistical mechanics, to find its partition function. Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the methods of multivariable calculus. That is, there is no elementary indefinite integral for but the definite integralcan be evaluated. The definite integral of an arbitrary Gaussian function is (Wikipedia).
Gaussian Integral 7 Wallis Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I calculate the Gaussian integral by using a technique that is very similar to the
From playlist Gaussian Integral
Gaussian Integral 8 Original Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I present the classical way using polar coordinates, the one that Laplace original
From playlist Gaussian Integral
Quantum Integral. Gauss would be proud! I calculate the integral of x^2n e^-x^2 from -infinity to infinity, using Feynman's technique, as well as the Gaussian integral and differentiation. This integral appears over and over again in quantum mechanics and is useful for calculus and physics
From playlist Integrals
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I calculate the Gaussian integral by using a Fubini-type argument, namely by calcu
From playlist Gaussian Integral
Gaussian Integral 6 Gamma Function
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I calculate the Gaussian integral by using properties of the gamma function, which
From playlist Gaussian Integral
Gaussian Integral 10 Fourier Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I show how the Gaussian integral appears in the Fourier transform: Namely if you t
From playlist Gaussian Integral
Gaussian Integral 11 Complex Analysis
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I use complex analysis to calculate the Gaussian integral. More precisely, I integ
From playlist Gaussian Integral
Video by vcubingx calculating the Gaussian integral: https://youtu.be/9CgOthUUdw4 This integral has a bunch of x and ln(x) in it...but Gauss is hiding in there after a little substitution! Check out more integrals here: https://www.youtube.com/playlist?list=PLug5ZIRrShJGoUgjNdl12Xs6j4YqT
From playlist Integrals
This is a follow-up of the integral of exp(-x^2) video on blackpenredpen's channel, in case you're wondering how to get that extra factor of r in the integral. It's mathemagical! :D Here's the like to the original video: Gaussian Integral https://youtu.be/r9W8YWELXvg
From playlist Double and Triple Integrals
Slides and more information: https://mml-book.github.io/slopes-expectations.html
From playlist There and Back Again: A Tale of Slopes and Expectations (NeurIPS-2020 Tutorial)
From playlist COMP0168 (2020/21)
Orli Herscovici - Kohler-Jobin Meets Ehrhard - IPAM at UCLA
Recorded 07 February 2022. Orli Herscovici of the Georgia Institute of Technology presents "Kohler-Jobin Meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements" at IPAM's Calculus of Variations in Probab
From playlist Workshop: Calculus of Variations in Probability and Geometry
Joe Neeman: Gaussian isoperimetry and related topics I
The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence
From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability
ML Tutorial: Probabilistic Numerical Methods (Jon Cockayne)
Machine Learning Tutorial at Imperial College London: Probabilistic Numerical Methods Jon Cockayne (University of Warwick) February 22, 2017
From playlist Machine Learning Tutorials
ME565 Lecture 19: Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain
ME565 Lecture 19 Engineering Mathematics at the University of Washington Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain Notes: http://faculty.washington.edu/sbrunton/me565/pdf/L19.pdf Course Website: http://faculty.washington.edu/sbrunton/me565/ http://faculty.wash
From playlist Engineering Mathematics (UW ME564 and ME565)
From playlist COMP0168 (2020/21)
Slides and more information: https://mml-book.github.io/slopes-expectations.html
From playlist There and Back Again: A Tale of Slopes and Expectations (NeurIPS-2020 Tutorial)
PUSHING A GAUSSIAN TO THE LIMIT
Integrating a gaussian is everyones favorite party trick. But it can be used to describe something else. Link to gaussian integral: https://www.youtube.com/watch?v=mcar5MDMd_A Link to my Skype Tutoring site: dotsontutoring.simplybook.me or email dotsontutoring@gmail.com if you have ques
From playlist Math/Derivation Videos
In this video we discuss the Gaussian (AKA Normal) probability distribution function. We show how it relates to the error function (erf) and discuss how to use this distribution analytically and numerically (for example when analyzing real-life sensor data or performing simulation of stoc
From playlist Probability