Category: Mathematical identities

Sun's curious identity
In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:
Capelli's identity
In mathematics, Capelli's identity, named after Alfredo Capelli, is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation th
Picone identity
In the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. Since its inception
Chain rule (probability)
In probability theory, the chain rule (also called the general product rule) permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabi
Pfister's sixteen-square identity
In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister aroun
Fierz identity
In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinor
Green's identities
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathem
List of set identities and relations
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms,i is the imaginary unit, which by definition satisfies i2 =
Q-Vandermonde identity
In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that
Power rule
In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomial
Noether identities
In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whos
Squared triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, The same equation may be written more compactly using the mathematical notation for summation: This
Dyson conjecture
In mathematics, the Dyson conjecture (Freeman Dyson ) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it
Vandermonde's identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre
Quintuple product identity
In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson and rediscovered by and . It is analogous to the Jacobi triple product identity, and is the Ma
Triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scal
Weitzenböck identity
In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two sec
Hockey-stick identity
In combinatorial mathematics, the identity or equivalently, the mirror-image by the substitution : is known as the hockey-stick, Christmas stocking identity, boomerang identity, or Chu's Theorem. The
Heine's identity
In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as where is a Legendre function of the second kin
Lagrange's identity (boundary value problem)
In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration
Dixon's identity
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of thr
Brahmagupta–Fibonacci identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under
Lists of integrals
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler c
General Leibniz rule
In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are -times differentiable fu
Cassini and Catalan identities
Cassini's identity (sometimes called Simson's identity) and Catalan's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of Catalan's identity, states t
List of trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined.
Mingarelli identity
In the field of ordinary differential equations, the Mingarelli identity is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations
Hypergeometric identity
In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in soluti
Bochner–Kodaira–Nakano identity
In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a
Lerche–Newberger sum rule
The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain infinite series involving Bessel functions Jα of the first kind. It states that if μ is an
Vector algebra relations
The following are important identities in vector algebra. Identities that involve the magnitude of a vector , or the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension.
Brahmagupta's identity
In algebra, Brahmagupta's identity says that, for given , the product of two numbers of the form is itself a number of that form. In other words, the set of such numbers is closed under multiplication
Jacobi–Anger expansion
In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example,
Exterior calculus identities
This article summarizes several identities in exterior calculus.
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the pro
Candido's identity
Candido's identity, named after the Italian mathematician Giacomo Candido, is an identity for real numbers. It states that for two arbitrary real numbers and the following equality holds: The identity
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Hartley kernel
No description available.
Weyl expansion
In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In Cartesian coordinate syst
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials
Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.
Enumerator polynomial
In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight. Let be a binary linear code length . The weight distribution i
Selberg's identity
In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by Selberg. Selberg and Erdős both used this identity to give elementary proofs of the prime number
Siegel identity
In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.
Bochner identity
In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician
Implicit function theorem
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by represe
Liouville's formula
In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous
Maximum-minimums identity
In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2n − 1 non-empty subsets of S. Let S = {x1, x2, ..., xn}. The iden
Vector calculus identities
The following are important identities involving derivatives and integrals in vector calculus.
Pascal's rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers n and k, where is a binomial coefficient; one i
Vaughan's identity
In mathematics and analytic number theory, Vaughan's identity is an identity found by R. C. Vaughan that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate sum
Morrie's law
Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it du
Cis (mathematics)
cis is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. The notation is less commonly used in mathema
Abel's identity
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear
Rogers–Ramanujan identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Roger
Identity (mathematics)
In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value fo
Fay's trisecant identity
In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by Fay . Fay's identity holds for theta functions of Jacobians of curves, but not
Tangent half-angle formula
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the
Degen's eight-square identity
In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.Namely: First discovered by Carl
Binet–Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that for every choice of real or complex numbers (or more generally, elements of a com
Pokhozhaev's identity
Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev and
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operat
List of logarithmic identities
In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.
Difference of two squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity i
Macdonald identities
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald. They include as special cases the Jacobi triple product id
Rothe–Hagen identity
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers except where its denominators vanish: It is a generalization of Vandermonde's identity, and is named a
Cas (mathematics)
No description available.
Rogers–Ramanujan continued fraction
The Rogers–Ramanujan continued fraction is a continued fraction discovered by and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated expl
Cyclotomic identity
In mathematics, the cyclotomic identity states that where M is Moreau's necklace-counting function, and μ is the classic Möbius function of number theory. The name comes from the denominator, 1 − z j,
Differentiation rules
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity: for complex numbers x and y, with |x| < 1 and y ≠ 0. It was introduced by Jacobi in his work Fundamenta Nova Theoriae Functionum
Pythagorean trigonometric identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles
Differentiation of trigonometric functions
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the der
Sommerfeld identity
The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves, where is to be taken with positive real part, to ensure the convergence of the in
Lagrange's identity
In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is: which applies to any two sets {a1, a2, ..., an} and {b1, b2, ..., bn} of real or complex numbers (or more generally, elements of
Hermite's identity
In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n th
List of mathematical identities
This article lists mathematical identities, that is, identically true relations holding in mathematics. * Bézout's identity (despite its usual name, it is not, properly speaking, an identity) * Bino
List of vector calculus identities
No description available.