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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) * Binet-cauchy 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 * List of logarithmic 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 ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (mathematics)
In mathematics, an identity is an equality (mathematics), equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variable (mathematics), variables) produce the same value for all values of the variables within a certain domain of discourse. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same function (mathematics), 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. Formally, an identity is a universally quantified equality. 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), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Trivial identities ''Trivial'' mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows: Explanations By definition, we know that: \log_b(y) = x \iff b^x = y, where b \neq 0 and b \neq 1. Setting x = 0, we can see that: b^x = y \iff b^ = y \iff 1 = y \iff y = 1 So, substituting these values into the formula, we see that: \log_b (y) = x \iff \log_b (1) = 0, which gets us the first property. Setting x = 1, we can see that: b^x = y \iff b^ = y \iff b = y \iff y = b So, substituting these values into the formula, we see that: \log_b (y) = x \iff \log_b (b) = 1, which gets us the second property. Cancelling exponentials Logarithms and exponentials with the same base cancel each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 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, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Calculus Identities
This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry. 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 ve ... [...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. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However ... [...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. ... [...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 A hat is a Headgear, head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorpor .../ ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain Identity
In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that \begin x^4 + 4y^4 &= \bigl((x + y)^2 + y^2\bigr)\cdot\bigl((x - y)^2 + y^2\bigr)\\ &= (x^2 + 2xy + 2y^2)\cdot(x^2 - 2xy + 2y^2). \end Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form x^4+4y^4, and it frequently forms the basis of problems in mathematics competitions. History Although the identity has been attributed to Sophie Germain, it does not appear in her works. Instead, in her works one can find the related identity \begin x^4+y^4 &= (x^2-y^2)^2+2(xy)^2\\ &= (x^2+y^2)^2-2(xy)^2.\\ \end Modifying this equation by multiplying y by \sqrt2 gives x^4+4y^4 = (x^2+2y^2)^2-4(xy)^2, a difference of two squares, from which Germain's identity follows. The inaccurate attribution of this identity to Germain was made by Leonard Eugene Dickson in his ''History of the Theory of Numbers'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sherman–Morrison Formula
In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed. That is, given an invertible matrix A and the outer product u v^\textsf of vectors u and v, the formula cheaply computes an updated matrix inverse \left(A + uv^\textsf\right)\vphantom)^. 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 if and only if 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 \Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfister's Sixteen-square Identity
In algebra, Pfister's sixteen-square identity is a non- bilinear identity of form \left(x_1^2+x_2^2+x_3^2+\cdots+x_^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_^2 It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is \begin &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \end If all x_i and y_i with i>8 are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The u_i are \begin &u_1 = \tfrac \\ &u_2 = \tfrac \\ &u_3 = \tfrac \\ &u_4 = \tfrac \\ &u_5 = \tfrac \\ &u_6 = \tfrac \\ &u_7 = \tfrac \\ &u_8 = \tfrac \end and, a=-1,\;\;b=0,\;\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parseval's Identity
In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes). Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors). The identity asserts that the sum of squares of the Fourier coefficients of a function is equal to the integral of the square of the function, \Vert f \Vert^2_ = \frac1\int_^\pi , f(x), ^2 \, dx = \sum_^\infty , \hat f(n), ^2, where the Fourier coefficients \hat f(n) of f are given by \hat f(n) = \frac \int_^ f(x) e^ \, dx. The result holds as stated, provided f is a square-integrable function or, more generally, in ''L''''p'' space L^2 \pi, \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
