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Sobolev Space

The Sobolev space W^k, p(U) consists of all locally summable functions u: U to R such that for each multi-index alpha with |alpha| <= k, the weak derivative D^alphau exists and belongs to L^p(U).

Laplace's Equation

Laplace's equation Delta u = 0 is the prototype elliptic PDE. Every major result in elliptic theory — mean value properties, maximum principles, regularity, Harnack inequalities — appears first and

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Maximum Modulus Principle

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be a bounded domain with closure $\overline{\Omega}$. A function $f: \overline{\Omega} \to \mathbb{C}$ is said to be holomorphic on $\overline{\Omega}$ (or: holomorphic on $\Omega$ and continuous on $\overline{\Omega}$) if: 1. $f$ is holomorphic on the open domain $\Omega$, meaning the complex derivative \begin{align} f'(z) &= \lim_{h \to 0} \frac{f(z+h) - f(z)}{h} \end{align} exists for every $z \in \Omega$. 2. $f$ is continuous on the closed domain $\overline{\Omega}$. We write $f \in \mathcal{O}(\Omega) \cap C(\overline{\Omega})$ to denote this class of functions.

Updated 2 days ago

Cauchy's Integral Formula

Analysis Complex Analysis

Let $\gamma: [a, b] \to \mathbb{C}$ be a closed piecewise $C^1$ curve and let $z_0 \in \mathbb{C} \setminus \gamma([a, b])$ be a point not on $\gamma$. The winding number of $\gamma$ around $z_0$ is \begin{align} n(\gamma, z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z - z_0}. \end{align}

Updated 2 days ago

Expectation

Probability & Statistics Probability Theory

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $X: \Omega \to \mathbb{R}$ be a random variable. Write $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$, so that $X = X^+ - X^-$ and $|X| = X^+ + X^-$. If $\min(\mathbb{E}[X^+], \mathbb{E}[X^-]) < \infty$, the expectation of $X$ is \begin{align} \mathbb{E}[X] = \mathbb{E}[X^+] - \mathbb{E}[X^-] \in [-\infty, \infty]. \end{align} The random variable $X$ is called integrable if $\mathbb{E}[|X|] = \mathbb{E}[X^+] + \mathbb{E}[X^-] < \infty$, in which case $\mathbb{E}[X] \in \mathbb{R}$.

Updated 2 days ago

Countable Set

Discrete Mathematics Set Theory

How big is infinity? The integers, the rationals, and the algebraic numbers are all infinite sets, yet every one of them can be arranged into a single list indexed by the natural numbers. The real numbers cannot. This distinction — between infinities that can be enumerated and those that resist enumeration — is the starting point of the theory of…

Updated 2 days ago

Entropy Solution

Analysis Partial Differential Equations

A pair of functions $(\eta, q)$ is called an entropy/entropy flux pair if: 1. $\eta: \mathbb{R} \to \mathbb{R}$ is a $C^2$ and convex function (the entropy). 2. $q: \mathbb{R} \to \mathbb{R}$ is a $C^1$ function (the entropy flux) that satisfies $q'(u) = \eta'(u) f'(u)$. If $u$ were a smooth solution, multiplying the PDE by $\eta'(u)$ would yield $\eta'(u) u_t + \eta'(u) f'(u) u_x = \partial_t \eta(u) + \partial_x q(u) = 0$. For weak solutions, this equality is relaxed to an inequality.

Updated 2 days ago

Functions of Bounded Variation

Analysis Real Analysis

Let $U \subseteq \mathbb{R}^n$ be an open set. A function $u \in L^1(U)$ is of bounded variation if its distributional derivative is representable by a finite vector-valued Radon measure. That is, $u \in BV(U)$ if there exists a finite $\mathbb{R}^n$-valued Radon measure $Du = (\partial_1 u, \ldots, \partial_n u)$ on $U$ such that for every $i \in \{1, \ldots, n\}$ and every test function $\phi \in C^\infty_c(U)$: \begin{align} \int_U u \, \partial_i \phi \, d\mathcal{L}^n = -\int_U \phi \, d(\partial_i u). \end{align} The right-hand side is integration against the signed Radon measure $\partial_i u$, not against an $L^1$ function.

Updated 2 days ago

Fejér Kernel

Analysis Harmonic Analysis

The Fejér kernel is the convolution kernel for the Cesàro means of Fourier series. Where the Dirichlet kernel represents the Fourier partial sums $S_N f$ and suffers from oscillation and growing $L^1$ norm, the Fejér kernel $F_N$ represents the averaged partial sums $\sigma_N f = \frac{1}{N+1}\sum_{n=0}^N S_n f$ and is everywhere non-negative.…

Updated 2 days ago

Cesàro Means

Analysis Harmonic Analysis

The Cesàro mean of a sequence $(s_n)$ is the arithmetic average $\sigma_N = \frac{1}{N+1}\sum_{n=0}^N s_n$. This is a summability method: a systematic way to assign a value to a sequence (or series) that may not converge in the ordinary sense. Cesàro summability is strictly weaker than ordinary convergence — every convergent sequence is…

Updated 2 days ago

Schwarz–Christoffel Mapping

Analysis Complex Analysis

The Riemann mapping theorem guarantees that any simply connected proper subdomain of $\mathbb{C}$ is conformally equivalent to the open unit disc $\mathbb{D}$ — but the proof, which proceeds via normal families and an extremal argument, is entirely non-constructive. It produces no formula for the mapping function, no algorithm for evaluating it,…

Updated 2 days ago

Convolution

Analysis Harmonic Analysis

Convolution is the operation that "smears" one function against another, producing a weighted average that inherits the best properties of both factors. It is the algebraic operation underlying three pillars of analysis: $L^p$ theory (where Young's inequality controls its integrability), Fourier analysis (where the Convolution Theorem converts it…

Updated 2 days ago

Laplace's Equation

Analysis Partial Differential Equations

Laplace's equation $\Delta u = 0$ is the prototype elliptic PDE. Every major result in elliptic theory — mean value properties, maximum principles, regularity, Harnack inequalities — appears first and most explicitly here, before being generalised to variable-coefficient operators on the second-order elliptic equations page. The Laplacian's full…

Updated 2 days ago

Heat Equation

Analysis Partial Differential Equations

The homogeneous Heat Equation for a function $u : \Omega \times (0, \infty) \to \mathbb{R}$ is the second-order linear PDE: \begin{align} \frac{\partial u}{\partial t} - \Delta u = 0 \end{align} where $\Omega \subseteq \mathbb{R}^n$ is an open set, $t > 0$ denotes time, and $\Delta = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$ is the spatial Laplacian.

Updated 2 days ago

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Maximum Modulus Principle

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Schwarz–Christoffel Mapping

Convolution

Laplace's Equation

Heat Equation

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Explore Mathematics

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Browse by Area

Trending This Week

Sobolev Space

Laplace's Equation

Table of content in different sections

Recently Updated

All Pages

Maximum Modulus Principle

Cauchy's Integral Formula

Expectation

Countable Set

Entropy Solution

Functions of Bounded Variation

Fejér Kernel

Cesàro Means

Schwarz–Christoffel Mapping

Convolution

Laplace's Equation

Heat Equation