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In [[mathematics]], the '''Struve functions''' {{math|'''H'''_''α''(''x'')}}, are solutions {{math|''y''(''x' The '''modified Struve functions''' {{math|'''L'''_''α''(''x'')}} are equal to {{math|−''ie''<sup> ...

...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. ...

{{Short description|Special function defined by an integral}} {{distinguish|text=[[List of integrals of exponential functions|other integrals]] of [[exponential function]]s}} ...

...in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the [[Tricomi–Carlitz polynomials]]. ...r equation]] for a one-electron atom. They also describe the static Wigner functions of oscillator systems in [[Phase space formulation#Simple harmonic oscillat ...

...the Appell double series entail [[recurrence relation]]s among contiguous functions. For example, a basic set of such relations for Appell's ''F''₁ ...in terms of [[multiple integral|double integral]]s involving [[elementary functions]] only {{harv|Gradshteyn|Ryzhik|2015|loc=§9.184}}. However, {{harvs | txt | ...

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

...ny particular [[special functions]] as special cases, such as [[elementary functions]], [[Bessel function]]s, and the [[orthogonal polynomials|classical orthogo ...irst3=O. I.|last3=Marichev|title=Integrals & Series Volume 3: More Special Functions|publisher=Gordon and Breach|year=1990|pages=439}}</ref> ...

{{Short description|Special function defined by an integral}} ...egral calculus]], an '''elliptic integral''' is one of a number of related functions defined as the value of certain integrals. Originally, they arose in conne ...

{{Short description|Special function defined by an integral}} ...urn from <math>(2\pi)</math> to 2 (at <math>t=2</math>). These alternative functions are usually known as '''normalized Fresnel integrals'''. ...

...= https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110 |title = Special functions of mathematics for engineers|page = 110|publisher = SPIE Press |date= 1998| This integral is a [[special function|special]] (non-[[elementary function|elementary]]) and [[sigmoid function|sigmoid]] function that occurs often in [[probabi ...

...are poles), but which are not generally solvable in terms of [[elementary functions]]. They were discovered by ...ers greater than 1, movable essential singularities can occur, and found a special case of what was later called Painleve VI equation (see below). ...

[[File:Vibrating drum Bessel function.gif|thumb|right|Bessel functions are the radial part of the modes of vibration of a circular drum.]] '''Bessel functions''', first defined by the mathematician [[Daniel Bernoulli]] and then genera ...

{{short description|Special mathematical function defined as sin(x)/x}} This Fourier integral, including the special case ...

...[function (mathematics)|function]] intended to include most of the known [[special function]]s as particular cases. This was not the only attempt of its kind: ...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 ...

...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. ...

...nt. {{math|''W''₀}} is known as the [[principal branch]]. These functions have the following property: if {{math|''z''}} and {{math|''w''}} are any c The Lambert {{mvar|W}} relation cannot be expressed in terms of [[elementary function]]s.<ref>{{citation ...

...are used more often in practical problems than the [[Weierstrass elliptic functions]] as they do not require notions of complex analysis to be defined and/or u ...denoted by pq(u,m), where p and q are any of the letters c, s, n, and d. (Functions of the form pp(u,m) are trivially set to unity for notational completeness. ...

...ns à une seule variable |trans-title=On the development of single-variable functions |journal=Bull. Acad. Sci. St. Petersb. |volume=1 |date=1859 |pages=193–200} ...form an [[orthonormal basis|orthogonal basis]] of the [[Hilbert space]] of functions satisfying ...

{{Short description|Special mathematical function}} ...''. Only for special values of ''s'' does the polylogarithm reduce to an [[elementary function]] such as the [[natural logarithm]] or [[rational function]]s. In ...

...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 ...anslated version of the factorial function,<ref>{{cite book |title=Special Functions: A Graduate Text |first1=Richard |last1=Beals |first2=Roderick |last2=Wong ...