In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the function F defined by the convolution of f with the Gaussian function The factor 1/√(4π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of F(x) one also writes W[f](x). Note that F(x) need not exist for every real number x, when the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function f describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod t = 1 time units later will be given by the function F. By using values of t different from 1, we can define the generalized Weierstrass transform of f. The generalized Weierstrass transform provides a means to approximate a given integrable function f arbitrarily well with analytic functions. (Wikipedia).
Infinite products & the Weierstrass factorization theorem
In this video we're going to explain the Weierstrass factorization theorem, giving rise to infinite product representations of functions. Classical examples are that of the Gamma function or the sine function. https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem https://en.wiki
From playlist Programming
Math 139 Fourier Analysis Lecture 18: Weierstrass approximation; Heat Equation on the Line
Fourier transform and convolutions; Plancherel's theorem. Weierstrass approximation theorem. Application to PDEs: time-dependent heat equation on the line. The heat kernel. Convolution of a Schwartz-class function with the heat kernel solves the heat equation.
From playlist Course 8: Fourier Analysis
Applying reimann sum for the midpoint rule and 3 partitions
👉 Learn how to approximate the integral of a function using the Reimann sum approximation. Reimann sum is an approximation of the area under a curve or between two curves by dividing it into multiple simple shapes like rectangles and trapezoids. In using the Reimann sum to approximate the
From playlist The Integral
understand Weierstrass substitution (the sneakiest integration technique)
Weierstrass Substitution and more on https://brilliant.org/blackpenredpen/ That link also gives you a 20% off discount on their annual premium subscription. Thanks for checking it out. 0:00 Weierstrass Substiutiton 0:16 is t=tan(x/2) obvious to you? 1:52 insight to the substitution 6:2
From playlist [Math For Fun] Brilliant Problems
Partialbruchzerlegung: Eine Einführung
Heute behandeln wir die Partialbruchzerlegung. Hierbei handelt es sich nur um eine kleine Einführung um die Verfahrensweise zu verstehen. An introduction to partial fraction decomposition - German version
From playlist Theorie und Beweise
Stephanie Chan, Integral points in families of elliptic curves
VaNTAGe Seminar, June 28, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Hindry-Silverman: https://eudml.org/doc/143604 Alpoge: https://arxiv.org/abs/1412.1047 Bhargava-Shankar: https://arxiv.org/abs/1312.7859 Brumer-McGuiness: https://www.ams.org/journal
From playlist Arithmetic Statistics II
Minimal Discriminants and Minimal Weiestrass Forms For Elliptic Curves
This goes over the basic invariants I'm going to need for Elliptic curves for Szpiro's Conjecture.
From playlist ABC Conjecture Introduction
Let's get Weierstrass: A SPICY INTEGRAL BOI USING WEIERSTRASS SUBSTITUTION!
Merch :v - https://teespring.com/de/stores/papaflammy Help me create more free content! =) https://www.patreon.com/mathable Dummy Variables: https://www.youtube.com/watch?v=SuW_fChaqIs Let us start off into this Random Week with a personal favourite of mine! :) I certainly know that the
From playlist Integrals
integral of 1/(2+cos(x)) , Weierstrass substitution
integral of 1/(2+cos(x)), weierstrass substitution, integral of 1/(2+cos(x)) a great way to integrate a rational expression that involves sin(x) and cos(x), check out my other videos for examples! blackpenredpen https://twitter.com/blackpenredpen
From playlist Weierstrass Substitution
weierstrass substitution, integral of 1/(1+sin(x)+cos(x))
weierstrass substitution, tangent half-angle substitution, integral of 1/(1+sin(x)+cos(x)), a great way to integrate a rational expression that involves sin(x) and cos(x), check out my other videos for examples! blackpenredpen https://twitter.com/blackpenredpen
From playlist Weierstrass Substitution
Analysis 1 - Convergent Subsequences: Oxford Mathematics 1st Year Student Lecture
This is the third lecture we're making available from Vicky Neale's Analysis 1 course for First Year Oxford Mathematics Students. Vicky writes: Does every sequence have a convergent subsequence? Definitely no, for example 1, 2, 3, 4, 5, 6, ... has no convergent subsequence. Does every b
From playlist Oxford Mathematics 1st Year Student Lectures
Abonniert den Kanal oder unterstützt ihn auf Steady: https://steadyhq.com/en/brightsideofmaths Ihr werdet direkt informiert, wenn ich einen Livestream anbiete. Hier erkläre ich kurz das Riemann-Integral mit Ober- und Untersumme. Die Definition ist übliche, die im 1. Semester eingeführt w
From playlist Analysis
An Arithmetic Refinement of Homological Mirror Symmetry for the 2-Torus - Yanki Lekili
Yanki Lekili University of Cambridge November 9, 2012 We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z[[q]]. It specializes to an equivalence, over Z, of the Fukaya category o
From playlist Mathematics
Riemann-Integral vs. Lebesgue-Integral
English version here: https://www.youtube.com/watch?v=PGPZ0P1PJfw Unterstützt den Kanal auf Steady: https://steadyhq.com/en/brightsideofmaths Ihr werdet direkt informiert, wenn ich einen Livestream anbiete. Hier erkläre ich den Unterschied zwischen Riemann-Integral und Lebesgue-Integral
From playlist Analysis
Math 101 Fall 2017 Bolzano Weierstrass for Sequences
Theorem: any accumulation point of a sequence is a subsequential limit. Theorem: (Bolzano-Weierstrass) Any bounded sequence of real numbers has a convergent subsequence.
From playlist Course 6: Introduction to Analysis (Fall 2017)
Math 131 120716 Ascoli-Arzela and Stone-Weierstrass
Theorem of Ascoli-Arzela. Stone-Weierstrass Theorem (density of polynomials in the space of continuous functions on a closed interval with respect to the supremum norm metric).
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis
Real Analysis - Part 10 - Bolzano-Weierstrass theorem
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From playlist Real Analysis
Bolzano-Weierstrass Theorem (Direct Proof) In this video, I present a more direct proof of the Bolzano-Weierstrass Theorem, that does not use any facts about monotone subsequences, and instead uses the definition of a supremum. This proof is taken from Real Mathematical Analysis by Pugh,
From playlist Sequences
How to use midpoint rienmann sum with a table
👉 Learn how to approximate the integral of a function using the Reimann sum approximation. Reimann sum is an approximation of the area under a curve or between two curves by dividing it into multiple simple shapes like rectangles and trapezoids. In using the Reimann sum to approximate the
From playlist The Integral