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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) * Binomial inverse theorem * Binomial identity * Brahmagupta–Fibonacci two-square identity * Candido's identity * Cassini and Catalan identities * Degen's eight-square identity * Difference of two squares * Euler's four-square identity * Euler's identity * Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities * Heine's identity * Hermite's identity * Lagrange's identity * Lagrange's trigonometric identities * MacWilliams identity * Matrix determinant lemma * Newton's identity * Parseval's identity * Pfister's sixteen-square identity * Sherman–Morrison formula * Sophie Germain identity * Sun's curious identity * Sylvester's determinant identity * Vandermonde's identity * Woodbury matrix identity Identities for classes of functions * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometric ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Determinant Lemma
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT. Statement Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that :\det\left(\mathbf + \mathbf^\textsf\right) = \left(1 + \mathbf^\textsf\mathbf^\mathbf\right)\,\det\left(\mathbf\right)\,. Here, uvT is the outer product of two vectors u and v. The theorem can also be stated in terms of the adjugate matrix of A: :\det\left(\mathbf + \mathbf^\textsf\right) = \det\left(\mathbf\right) + \mathbf^\textsf\mathrm\left(\mathbf\right)\mathbf\,, in which case it applies whether or not the square matrix A is invertible. Proof First the proof of the special case A = I follows from the equality: : \begin \mathbf & 0 \\ \mathbf^\textsf & 1 \end \begin \mathbf + \mathbf^\textsf & \mathbf \\ 0 & 1 \end \begin \mathbf & 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Logarithmic Functions
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. ''Note:'' ''x'' > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity. Integrals involving only logarithmic functions : \int\log_a x\,dx = x\log_a x - \frac = \frac : \int\ln(ax)\,dx = x\ln(ax) - x : \int\ln (ax + b)\,dx = \frac : \int (\ln x)^2\,dx = x(\ln x)^2 - 2x\ln x + 2x : \int (\ln x)^n\,dx = x\sum^_(-1)^ \frac(\ln x)^k : \int \frac = \ln, \ln x, + \ln x + \sum^\infty_\frac : \int \frac = \operatorname(x), the logarithmic integral. : \int \frac = -\frac + \frac\int\frac \qquad\mboxn\neq 1\mbox : \int \ln f(x)\,dx = x\ln f(x) - \int x\frac\,dx \qquad\mbox f(x) > 0\mbox Integrals involving logarithmic and power functions : \int x^m\ln x\,dx = x^\left(\frac-\frac\right) \qquad\mboxm\neq -1\mbox : \int x^m (\ln x)^n\,dx = \frac - \frac\int x^m (\ln x)^ dx \qquad\mbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci numbers include co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Calculus Identities
This article summarizes several identities in exterior calculus. Notation The following summarizes short definitions and notations that are used in this article. Manifold M, N are n-dimensional smooth manifolds, where n\in \mathbb . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page. p \in M , q \in N denote one point on each of the manifolds. The boundary of a manifold M is a manifold \partial M , which has dimension n - 1 . An orientation on M induces an orientation on \partial M . We usually denote a submanifold by \Sigma \subset M. Tangent and cotangent bundles TM, T^M denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold M. T_p M , T_q N denote the tangent spaces of M, N at the points p, q, respectively. T^_p M denotes the cotangent space of M at the point p. Sections of the tangent bundles, also known as vector fields, are typically denoted as X, Y, Z \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodbury Matrix Identity
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is : \left(A + UCV \right)^ = A^ - A^U \left(C^ + VA^U \right)^ VA^, where ''A'', ''U'', ''C'' and ''V'' are conformable matrices: ''A'' is ''n''×''n'', ''C'' is ''k''×''k'', ''U'' is ''n''×''k'', and ''V'' is ''k''×''n''. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. Discussion To prove this result, we will start by proving a simpler one. Replacing ''A'' and ''C'' with the ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde's Identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: :=\sum_^r for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie.See for the history. There is a ''q''-analog to this theorem called the ''q''-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity : = \sum_ \cdots . Proofs Algebraic proof In general, the product of two polynomials with degrees ''m'' and ''n'', respectively, is given by :\biggl(\sum_^m a_ix^i\biggr) \biggl(\sum_^n b_jx^j\biggr) = \sum_^\biggl(\sum_^r a_k b_\biggr) x^r, where we use the convention that ''ai'' = 0 for all integers ''i'' > ''m'' and ''bj'' = 0 for all integers ''j'' > ''n''. By the binomial theorem, :(1+x)^ = \sum_^ x^r. U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Determinant Identity
In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851. Cited in Given an ''n''-by-''n'' matrix A, let \det(A) denote its determinant. Choose a pair :u =(u_1, \dots, u_m), v =(v_1, \dots, v_m) \subset (1, \dots, n) of ''m''-element ordered subsets of (1, \dots, n), where ''m'' ≤ ''n''. Let A^u_v denote the (''n''−''m'')-by-(''n''−''m'') submatrix of A obtained by deleting the rows in u and the columns in v. Define the auxiliary ''m''-by-''m'' matrix \tilde^u_v whose elements are equal to the following determinants : (\tilde^u_v)_ := \det(A^_), where uhat/math>, vhat/math> denote the ''m''−1 element subsets of u and v obtained by deleting the elements u_i and v_j, respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851): :\det(A)(\det(A^u_v))^=\det(\tilde^u_v). When ''m'' =& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: : (x+m+1)\sum_^m(-1)^i\dbinom\dbinom -\sum_^\dbinom(-4)^i=(x-m)\dbinom. Proofs After Sun's publication of this identity in 2002, five other proofs were obtained by various mathematicians: * Panholzer and Prodinger's proof via generating functions; * Merlini and Sprugnoli's proof using Riordan arrays; * Ekhad and Mohammed's proof by the WZ method; * Chu and Claudio's proof with the help of Jensen's formula; * Callan's combinatorial proof involving dominos and colorings. References *. *. *. *. *. *. *{{citation , last = Sun , first = Zhi-Wei , doi = 10.1016/j.disc.2007.08.046 , arxiv = math.NT/0404385 , issue = 18 , journal = Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain Identity
Sophie is a version of the female given name Sophia (given name), Sophia, meaning "wise". People with the name Born in the Middle Ages * Sophie, Countess of Bar (c. 1004 or 1018–1093), sovereign Countess of Bar and lady of Mousson * Sophie of Thuringia, Duchess of Brabant (1224–1275), second wife and only Duchess consort of Henry II, Duke of Brabant and Lothier Born in 1600s and 1700s * Sophie of Anhalt-Zerbst (1729–1796), later Empress Catherine II of Russia * Sophie Amalie of Brunswick-Lüneburg (1628–1685), Queen consort of Denmark-Norway * Sophie Blanchard (1778–1819), French balloonist * Sophie Dorothea of Württemberg (1759–1828), second wife of Tsar Paul I of Russia * Sophie Dawes, Baronne de Feuchères ( 1795–1840), English baroness * Sophie Germain (1776–1831), French mathematician * Sophie Piper (1757–1816), Swedish countess * Sophie Schröder (1781–1868), German actress * Sophie von La Roche (1730–1807), German author Born 1790–1918 * Soph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sherman–Morrison Formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications. Statement Suppose A\in\mathbb^ is an invertible square matrix and u,v\in\mathbb^n are column vectors. Then A + uv^\textsf is invertible iff 1 + v^\textsf A^u \neq 0. In this case, :\left(A + uv^\textsf\right)^ = A^ - . Here, uv^\textsf is the outer product of two vectors u and v. The general form shown here is the one published by Bartlett. Proof (\Leftarrow) To prove that the backward direction 1 + v^\textsfA^u \neq 0 \Rightarrow A + uv^\textsf is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y (in this case the right-ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
