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...Legendre polynomials in the context of [[potential theory]] and [[harmonic analysis]]. The [[Newtonian potential]] in '''R'''^''n'' has the expansio ...t2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978 ...

...h>SL(2,\mathbb{R})</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by ...=Watson | first2=G. N. | authorlink2=G. N. Watson|title=A Course in Modern Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-58807-2 | year= ...

...37/0507003 | mr=0399537 | year=1976 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=7 | issue=1 | pages=16–28}} * Atakishiyeva, M. K., & Atakishiyev, N. M. (1997). Fourier-Gauss transforms of the Al-Salam-Chihara polynomials. Journal of Physics A: ...

The normalized sinc function is the [[Fourier transform]] of the [[rectangular function]] with no scaling. It is used in ...unication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",< ...

[[Asymptotic analysis|Asymptotic expansions]] for these limits are also available. These are list ==Fourier transform== ...

* [[Generalized Fourier series]] * {{cite book | first=Dunham |last=Jackson | title= Fourier Series and Orthogonal Polynomials | location= New York | publisher=Dover | ...

...h>. At <math>s = 1</math> it has a [[simple pole]] with [[residue (complex analysis)|residue]] <math>1</math>. The constant term is given by ==Fourier transform== ...

...''n'' and ''j'' = 1, 2, ..., ''m'', which implies that no [[pole (complex analysis)|pole]] of any Γ(''b''_''j'' − ''s''), ''j'' = 1, 2, ..., ''m'', ...then the Meijer G-function can be expressed as a sum of [[residue (complex analysis)|residue]]s in terms of [[generalized hypergeometric function]]s <sub>''p'' ...

* in [[numerical analysis]] as [[Gaussian quadrature]]; ...{{math|''F''(''it'') {{=}} 0}} for every real {{mvar|t}} means that the [[Fourier transform]] of {{math|''f''(''x'')''e''^−''x''²}} is 0 ...

...exp(2π''iz'')}}. It is a [[Jacobi form]]. At fixed {{mvar|τ}}, this is a [[Fourier series]] for a 1-periodic [[entire function]] of {{mvar|z}}. Accordingly, t ...theta-function and their applications | journal = Journal of Mathematical Analysis and Applications | pages = 381–400 | volume = 292 | issue = 2 | doi=10.101 ...

...the theory of [[random matrices]]), [[approximation theory]], [[numerical analysis]], and many others. ...tions (with eigenvalue (&minus;''i'' ^''n'') of the [[continuous Fourier transform]]. ...

...essed via the expansion:<ref name = boyd>{{cite book|title = Chebyshev and Fourier Spectral Methods|first = John P.|last = Boyd|isbn = 0-486-41183-4|edition = ...change of variables, all of the theorems, identities, etc. that apply to [[Fourier series]] have a Chebyshev counterpart.<ref name=boyd/> These attributes inc ...

...ity]]. They tend to occur in problems involving periodic motion, or in the analysis of [[partial differential equation]] [[boundary value problem]]s possessing Mathieu functions of the first kind can be represented as [[Fourier series]]:<ref name="Arscott_chapIII"/> ...

...arithm and the orthogonality of the exponential terms (see e.g. [[discrete Fourier transform]]). ...h complex [[roots of unity]] are given by the [[discrete Fourier transform|Fourier sum]]: ...

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example: The Bessel functions have the following [[Asymptotic analysis|asymptotic]] forms. For small arguments {{math|0 < ''z'' ≪ {{sqrt|''α'' + 1 ...

....ii DLMF, Incomplete Gamma functions, analytic continuation]</ref> Complex analysis shows how properties of the real incomplete gamma functions extend to their ...ss]], here developed for ''s'' → 0, fills in the missing values. [[Complex analysis]] guarantees [[holomorphic function|holomorphicity]], because <math>\Gamma( ...

...rier series]]. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see [[spher ...'', Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 1976, Vol. 7, No. 1 : pp.&nbsp;59–69 ...

...math>z</math>, at a [[simple pole]] <math>c</math>, the [[Residue (complex analysis)|residue]] of <math>f</math> is given by: === Fourier series expansion === ...