Knowledge Base

Explore Mathematics

Articles, definitions, and theorems — organized by area

298 pages 4830 theorems 4 contributors 88 edits this week

Subareas

Create New Bundle

Bundle Name

Browse by Area

View all areas →

Trending This Week

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

Analysis 0

Table of content in different sections

Topic 1 Subtopic 1.1 And so on asd Subtopi 2 asd Topic 2 asd Ad

Discrete Mathematics 0

All Pages

Lebesgue Integral

Analysis Measure Theory

The Lebesgue integral is the modern theory of integration, replacing the Riemann integral's strategy of partitioning the domain with a fundamentally different approach: partitioning the range. Given a function $f$ on a measure space $(E, \mathcal{E}, \mu)$, the Lebesgue integral asks not "what does $f$ do on each subinterval?" but "on what set…

Updated 2 days ago

Hilbert Transform

Analysis Harmonic Analysis

The Hilbert transform is the prototypical singular integral operator on the real line. Given a function $f : \mathbb{R} \to \mathbb{C}$, the Hilbert transform $Hf$ is defined — at least formally — by the convolution \begin{align} Hf(x) = \frac{1}{\pi} \int_{\mathbb{R}} \frac{f(t)}{x - t} \, d\mathcal{L}^1(t). \end{align} The integral does not…

Updated 2 days ago

Cambridge IB Probability and Measure

Probability & Statistics Probability Theory

The central question of this course is deceptively simple: when can we interchange a limit and an integral? That is, when does \begin{align} \lim_{n \to \infty} \int f_n \, d\mu = \int \lim_{n \to \infty} f_n \, d\mu \end{align} hold? In the Riemann theory taught in Analysis II, the answer is unsatisfying: only under strong hypotheses (typically…

Updated 2 days ago

Inhomogeneous Sobolev Space

Let $s \in \mathbb{R}$ and $n \ge 1$. The inhomogeneous Sobolev space of order $s$ is \begin{align} H^s(\mathbb{R}^n) &:= \left\{u \in \mathcal{S}'(\mathbb{R}^n) \;\middle|\; (1+|\xi|^2)^{s/2}\hat{u} = T_g \text{ for some } g \in L^2(\mathbb{R}^n)\right\}, \end{align} where $(1+|\xi|^2)^{s/2}\hat{u}$ denotes the product of the slowly increasing function $(1+|\xi|^2)^{s/2} \in \mathcal{O}_M(\mathbb{R}^n)$ with the tempered distribution $\hat{u} \in \mathcal{S}'(\mathbb{R}^n)$, acting on test functions by \begin{align} \bigl((1+|\xi|^2)^{s/2}\hat{u}\bigr)(\phi) &= \hat{u}\bigl((1+|\xi|^2)^{s/2}\phi\bigr) \quad \text{for every } \phi \in \mathcal{S}(\mathbb{R}^n), \end{align} and $T_g \in \mathcal{S}'(\mathbb{R}^n)$ denotes the regular distribution associated to $g$ via \begin{align} T_g(\phi) &:= \int_{\mathbb{R}^n} g(\xi)\,\phi(\xi) \, d\mathcal{L}^n(\xi). \end{align} The norm is $\|u\|_{H^s} := \|g\|_{L^2(\mathbb{R}^n)}$.

Updated 2 days ago

Homogeneous Sobolev Space

Analysis Partial Differential Equations

The space of tempered distributions modulo polynomials, denoted $\mathcal{S}'(\mathbb{R}^n)/\mathcal{P}$, is the quotient of $\mathcal{S}'(\mathbb{R}^n)$ by the subspace $\mathcal{P}$ of all polynomials on $\mathbb{R}^n$. Two tempered distributions $f, g \in \mathcal{S}'(\mathbb{R}^n)$ are identified in this quotient if and only if $f - g \in \mathcal{P}$.

Updated 2 days ago

Weak Convergence

Analysis Functional Analysis

Let $X$ be a real Banach space with continuous dual $X^$, consisting of all bounded linear functionals: \begin{align} f: X &\to \mathbb{R} \\ x &\mapsto f(x). \end{align} The weak topology on $X$ is the coarsest topology under which every evaluation map $x \mapsto f(x)$ is continuous, for all $f \in X^$.

Updated 2 days ago

Laurent Series

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be an open set, $z_0 \in \mathbb{C}$, and suppose $f: \Omega \to \mathbb{C}$ is holomorphic on the annulus $A(z_0; r, R) \subset \Omega$. The Laurent series of $f$ centered at $z_0$ on $A(z_0; r, R)$ is the bi-infinite series \begin{align} \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, \end{align} where the coefficients $a_n \in \mathbb{C}$ are given by the formula \begin{align} a_n &= \frac{1}{2\pi i} \oint_{\gamma} \frac{f(w)}{(w - z_0)^{n+1}}\, dw \end{align} for any positively oriented simple closed curve $\gamma$ in $A(z_0; r, R)$ with $z_0$ in its interior. The series converges absolutely and uniformly on compact subsets of $A(z_0; r, R)$.

Updated 2 days ago

Meromorphic Function

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be a connected open set. A function $f: \Omega \to \mathbb{C} \cup \{\infty\}$ is meromorphic on $\Omega$ if there exists a discrete closed subset $P \subset \Omega$ such that: 1. $f$ is holomorphic on $\Omega \setminus P$, 2. every point of $P$ is a pole of $f$. The set $P$ is called the pole set of $f$. We write $f \in \mathcal{M}(\Omega)$ for the class of meromorphic functions on $\Omega$.

Updated 2 days ago

Analytic Continuation

Analysis Complex Analysis

Let $\Omega_1, \Omega_2 \subset \mathbb{C}$ be connected open sets with $\Omega_1 \cap \Omega_2 \neq \varnothing$, and let $f_j: \Omega_j \to \mathbb{C}$ be holomorphic for $j = 1, 2$. We say that $f_2$ is a direct analytic continuation of $f_1$ to $\Omega_2$ if \begin{align} f_1(z) &= f_2(z) \quad \text{for all } z \in \Omega_1 \cap \Omega_2. \end{align}

Updated 2 days ago

Winding Number

Analysis Complex Analysis

Let $\gamma: [a, b] \to \mathbb{C}$ be a piecewise $C^1$ closed curve, so that $\gamma(a) = \gamma(b)$. Let $z_0 \in \mathbb{C} \setminus \operatorname{im}(\gamma)$. The winding number of $\gamma$ around $z_0$ is the integer \begin{align} n(\gamma, z_0) &= \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z - z_0}. \end{align}

Updated 2 days ago

Argument Principle

Analysis Complex Analysis

Let $\Omega \subset \mathbb{C}$ be an open set. A function $f: \Omega \setminus S \to \mathbb{C}$ is meromorphic on $\Omega$ if $S \subset \Omega$ is a discrete set (no accumulation points in $\Omega$), $f$ is holomorphic on $\Omega \setminus S$, and every point $z_0 \in S$ is a pole of $f$: there exists $m \in \mathbb{N}$ and a holomorphic function $g$ near $z_0$ with $g(z_0) \neq 0$ such that \begin{align} f(z) = \frac{g(z)}{(z - z_0)^m} \end{align} near $z_0$. The integer $m$ is the order of the pole at $z_0$, written $\operatorname{ord}(f, z_0) = -m$.

Updated 2 days ago

Liouville's Theorem

Analysis Complex Analysis

A function $f: \mathbb{C} \to \mathbb{C}$ is entire if it is holomorphic at every point of $\mathbb{C}$, i.e., if $f \in \mathcal{O}(\mathbb{C})$.

Updated 2 days ago

Showing 12 of 298 pages

What brings you to Androma?

Start with a route through the knowledge graph.

Explore Mathematics

Browse by Area

Trending This Week

Recently Updated

All Pages

Lebesgue Integral

Hilbert Transform

Cambridge IB Probability and Measure

Inhomogeneous Sobolev Space

Homogeneous Sobolev Space

Weak Convergence

Laurent Series

Meromorphic Function

Analytic Continuation

Winding Number

Argument Principle

Liouville's Theorem

Sign in to Androma

Check your inbox

One last step

Explore Mathematics

Create New Bundle

Browse by Area

Trending This Week

Sobolev Space

Laplace's Equation

Table of content in different sections

Recently Updated

All Pages

Lebesgue Integral

Hilbert Transform

Cambridge IB Probability and Measure

Inhomogeneous Sobolev Space

Homogeneous Sobolev Space

Weak Convergence

Laurent Series

Meromorphic Function

Analytic Continuation

Winding Number

Argument Principle

Liouville's Theorem