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List Of Mathematical Series
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. *Here, 0^0 is taken to have the value 1 *\ denotes the fractional part of x *B_n(x) is a Bernoulli polynomial. *B_n is a Bernoulli number, and here, B_1=-\frac. *E_n is an Euler number. *\zeta(s) is the Riemann zeta function. *\Gamma(z) is the gamma function. *\psi_n(z) is a polygamma function. *\operatorname_s(z) is a polylogarithm. * n \choose k is binomial coefficient *\exp(x) denotes exponential of x Sums of powers See Faulhaber's formula. *\sum_^m k^=\frac The first few values are: *\sum_^m k=\frac *\sum_^m k^2=\frac=\frac+\frac+\frac *\sum_^m k^3 =\left frac\right2=\frac+\frac+\frac See zeta constants. *\zeta(2n)=\sum^_ \frac=(-1)^ \frac The first few values are: *\zeta(2)=\sum^_ \frac=\frac (the Basel problem) *\zeta(4)=\sum^_ \frac=\frac *\zeta(6)=\sum^_ \frac=\frac Power series Low-order polylogarithms Finite sums: *\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero To The Power Of Zero
Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines . In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression. Discrete exponents Many widely used formulas involving natural-number exponents require to be defined as . For example, the following three interpretations of make just as much sense for as they do for positive integers : * The interpretation of as an empty product assigns it the value . * The combinatorial interpretation of is the number of 0-tuples of elements from a -element set; there is exactly one 0-tuple. * The set-theoretic interpretation of is the number of functions from the empty set to a -element set; there is exactly one such function, namely, the empty function. All three of these specialize t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Touchard Polynomials
The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by :T_n(x)=\sum_^n S(n,k)x^k=\sum_^n \left\x^k, where S(n,k)=\left\is a Stirling number of the second kind, i.e., the number of partitions of a set of size ''n'' into ''k'' disjoint non-empty subsets. Properties Basic properties The value at 1 of the ''n''th Touchard polynomial is the ''n''th Bell number, i.e., the number of partitions of a set of size ''n'': :T_n(1)=B_n. If ''X'' is a random variable with a Poisson distribution with expected value λ, then its ''n''th moment is E(''X''''n'') = ''T''''n''(λ), leading to the definition: :T_(x)=e^\sum_^\infty \frac . Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities: :T_n(\lambda+\mu)=\sum_^n T_k(\lambda) T_(\mu). The Touchard polynomials constitute the only polynomial sequence o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted simply as \sin \theta and \cos \theta. More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and lig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde 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. Usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Binomial Coefficient
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at ''n'' = 0 are: :, , , , , , 924, 3432, 12870, 48620, ...; Properties The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects. For example, n=2 represents ''AABB, ABAB, ABBA, BAAB, BABA, BBAA''. They also represent the number of combinations of ''A'' and ''B'' where there are never more ''B'' 's than ''A'' 's. For example, n=2 represents ''AAAA, AAAB, AABA, AABB, ABAA, ABAB''. The number of factors of ''2'' in \binom is equal to the number of ones in the binary representation of ''n'', so ''1'' is the only odd central binomial coefficient. Generating function The ordinary generating fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Numbers
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haversine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'', Section I) s. The versine of an angle is 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the of navigation. Overview The versine[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
