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...] and ''s'' a [[complex variable]] with [[real part]] greater than 1. By [[analytic continuation]], this function can be extended to a [[meromorphic function]] These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in { ...

==Analytic continuation== ...] <math>C</math> is a loop around the negative real axis. This provides an analytic continuation of <math>\zeta (s,q)</math>. ...

...ther". There are several common standard forms of confluent hypergeometric functions: ...ns are essentially the same, and differ from each other only by elementary functions and change of variables. ...

...icsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/ | title=The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function ...

...urn from <math>(2\pi)</math> to 2 (at <math>t=2</math>). These alternative functions are usually known as '''normalized Fresnel integrals'''. ...can be extended to the domain of [[complex number]]s, where they become [[analytic function]]s of a complex variable. ...

...ny particular [[special functions]] as special cases, such as [[elementary functions]], [[Bessel function]]s, and the [[orthogonal polynomials|classical orthogo ...en the series defines an [[analytic function]]. Such a function, and its [[analytic continuation]]s, is called the '''hypergeometric function'''. ...

...le integrals of trigonometric functions|List of integrals of trigonometric functions}} ...[[sinc function]], and also the zeroth [[Bessel function#Spherical Bessel functions: jn.2C yn|spherical Bessel function]]. ...

...upper''' and '''lower incomplete gamma functions''' are types of [[special functions]] which arise as solutions to various mathematical problems such as certain ...]</ref> Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts. ...

...939-02-06649-2.pdf A q-series identity and the Arithmetic of Hurwitz Zeta Functions]'', (2003) Proceedings of the [[American Mathematical Society]] '''131''', ...l for _''r''+1φ_''r''. This contour integral gives an analytic continuation of the basic hypergeometric function in ''z''. ...

...ood to be the limit value 1. The sinc function is then [[Analytic function|analytic]] everywhere and hence an [[entire function]]. ...nterpolation]] of [[sampling (signal processing)|sampled]] [[bandlimited]] functions: ...

...eneralize the last formula into a bivariate identity for a product of beta functions: ...l converges for all values of {{mvar|α}} and {{mvar|β}} and so gives the [[analytic continuation]] of the beta function. ...

...lues of ''x'' and ''y'' the function ''F''₁ can be defined by [[analytic continuation]]. It can be shown<ref>See Burchnall & Chaundy (1940), formula ...the Appell double series entail [[recurrence relation]]s among contiguous functions. For example, a basic set of such relations for Appell's ''F''₁ ...

...ers]] and [[binomial coefficient]]s. They are used for series expansion of functions, and with the [[Euler–MacLaurin formula]]. ...h, when appropriately scaled, the [[trigonometric function|sine and cosine functions]]. ...

...able property is the [[closure (mathematics)|closure]] of the set of all G-functions not only under differentiation but also under indefinite integration. In co ...''cx''^''γ'')·''G''₂(''dx''^''δ'') of two G-functions with [[rational number|rational]] ''γ''/''δ'' equals just another G-functio ...

...s to the [[generalized hypergeometric function]]. For other hypergeometric functions see [[#See also|See also]].}} ...ented by the '''hypergeometric series''', that includes many other special functions as [[special case|specific]] or [[limiting case (mathematics)|limiting case ...

{{short description|Special functions of several complex variables}} {{for|other θ functions|Theta function (disambiguation)}} ...

{{distinguish|text=[[List of integrals of exponential functions|other integrals]] of [[exponential function]]s}} ...taken on the negative real axis and ''E''₁ can be defined by [[analytic continuation]] elsewhere on the complex plane. ...

In [[mathematics]], '''Mathieu functions''', sometimes called angular Mathieu functions, are solutions of Mathieu's [[differential equation]] === Mathieu functions === ...

The gamma function then is defined as the [[analytic continuation]] of this integral function to a [[meromorphic function]] that ...popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of [[probability]] and [[statis ...

{{short description|Analytic function}} ...[[function (mathematics)|function]] of a [[complex variable]] ''s'' that [[analytic continuation|analytically continues]] the sum of the [[Dirichlet series]] ...