Mathematical analysis

branch of mathematics

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a d

mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function

set equipped with a metric (distance function)

right|thumb|250px|The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) right|thumb|250px|A curve intersecting an asymptote infinitely many times

largest and smallest value taken by a function in a given range

a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics

expression of a rational as an iterative sequence of addition and inversion of integers

the result yielded by a real number when divided by zero

set in a topological space that contains an open superset of a given point or subset

thumb|upright|Underdamped spring–mass system with

property limiting the "growth" of distances of outputs of a function uniformly across its domain

type of average

set is called bounded, if it is, in a certain sense, of finite size

element of a nonstandard model of the reals, which can be infinite or infinitesimal

in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability

discrete analog of a derivative

mathematical operation

function whose range is a subset of the real numbers

expression representing the product of an infinite sequence

construct related to weighted sums and averages

point at which a function is not continuous

derivative of a function of several variables with respect to one variable, without the others held constant

value that can be assigned to certain divergent integrals over a finite interval

linear combination of points whose coefficients are non-negative and sum to one

thumb|right|An upper semicontinuous function that is not lower semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right). thumb|right|A lower semicontinuous function that is not upper semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right).

technique in mathematics, statistics, and computer science to make a model more generalizable and transferable

mathematical expression with no necessarily obvious value

increasing sequence of numbers that span an interval

American award for mathematical analysis

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

a natural number whose square "ends" in the same digits as the number itself

series which gives an approximation to a function as the argument tends to some point

series in which the summands are not just real or complex numbers but functions

number of times an element appears in the multiset

the set of points lying on or above the graph of a function

thumb|The imaginary part of the complex logarithm. Trying to define the complex logarithm on \C-\{0\} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of \C-\{0\} by a [[helicoid (an example of a Riemann surface).]]

concept in mathematics

Concept in mathematical analysis

function whose squared absolute value has finite integral

two-volume publication by Leonhard Euler

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions.

decomposition of a positive real number into a series of unit fractions, each an integer multiple of the next one

connected open subset of a finite-dimensional vector space

measure of the local oscillation behavior of a function

function whose domain a subset of real numbers

summability method in physics

polynomial in the logarithm of n

Order-independent convergence of a sequence

mathematical concept

statistic

mathematical function

basic principle of asymptotic analysis due to George Gabriel Stokes and Lord Kelvin

trying to map moments to a measure that generates them

Property of functions

linear combination where the coefficients are non-negative

type of point where a function does not oscillate too much, roughly

Mathematical analysis term

notation used to describe discontinuous functions