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...articular, according to the [[Siegel-Walfisz theorem]] it is a very good [[approximation]] to the [[prime-counting function]], which is defined as the number of [[p ...on, this series is [[divergent series|not convergent]]: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large ...

...'-polynomials for non-standard parameters|journal=Journal of Approximation Theory|date=September 2011|volume=163|issue=9|pages=1246–1268|doi=10.1016/j.jat.20 ...

...r the [[normal distribution]]. The [[characteristic function (probability theory)|characteristic function]] for the (centered) Voigt profile will then be th === Pseudo-Voigt approximation === ...

...'-polynomials for non-standard parameters|journal=Journal of Approximation Theory|date=September 2011|volume=163|issue=9|pages=1246–1268|doi=10.1016/j.jat.20 ...

==Approximation== [[Stirling's approximation]] gives the asymptotic formula ...

*[[Sturm–Liouville theory|Sturm-Liouville theory]] ...d R. Kumar, Analysis of rhythms of EEG signals using orthogonal polynomial approximation, ACM International Conference on Convergence and Hybrid Information Technol ...

...mptoticExpansionE1.png|right|200px|thumb| Relative error of the asymptotic approximation for different number <math>~N~</math> of terms in the truncated sum]] ...stein and Handelsman, p.&nbsp;2</ref> However, there is a divergent series approximation that can be obtained by integrating <math>z e^z E_1(z)</math> by parts:<ref ...

...under the [[maximum norm]], also called the "[[minimax]]" criterion. This approximation leads directly to the method of [[Clenshaw–Curtis quadrature]]. The generating function relevant for 2-dimensional [[potential theory]] and [[Cylindrical multipole moments|multipole expansion]] is ...

...account of its connection with "the theory of Probability, and notably the theory of [[errors and residuals|Errors]]."<ref name="Glaisher1871a">{{cite journa ...ac{31}{200}</math> and <math>c_2 = -\frac{341}{8000},</math> the resulting approximation shows its largest relative error at <math>x=\pm 1.3796,</math> where it is ...

...hysics (in particular, the theory of [[random matrices]]), [[approximation theory]], [[numerical analysis]], and many others. This is a [[Sturm–Liouville theory|Sturm–Liouville]] type of equation. Such equations generally have singular ...

...ide uniform semiclassical approximations near a turning point in the [[WKB approximation]], when the potential may be locally approximated by a linear function of p ...symptotic because the ratio between Ai(−''z'') or Bi(−''z'') and the above approximation goes to infinity whenever the sine or cosine goes to zero. ...

...ner_D-matrix#Wigner_(small)_d-matrix|Wigner function]]. Generally a better approximation obeying the Regge symmetry is given by The following quantity acts as a [[metric tensor]] in angular-momentum theory and is also known as a ''Wigner 1-jm symbol'':<ref name="Wigner1951"/> ...

* in [[systems theory]] in connection with nonlinear operations on [[Gaussian noise]]. * in [[random matrix theory]] in Wigner–Dyson ensembles. ...

..., to unwind it to an approximation for <math>z!</math>. Furthermore, this approximation is exact in the limit as <math>n</math> goes to infinity. ...e [[Riemann zeta function]], and <math>\pi</math> is a [[Partition (number theory)|partition]] of <math>n</math> given by ...

...so be expressed in [[Carlson symmetric form]]. Additional insight into the theory of the elliptic integral may be gained through the study of the [[Schwarz–C This approximation has a relative precision better than {{val|3|e=−4}} for {{math|''k'' < {{sf ...

==Approximation in terms of hyperbolic functions== * [[Naum Akhiezer|N. I. Akhiezer]], ''Elements of the Theory of Elliptic Functions'' (1970) Moscow, translated into English as ''AMS Tra ...

...can draw an ellipse using an [[Ellipse#Approximation by osculating circles|approximation by the four osculating circles at the vertices]]. ...re based on the second paperstrip method.[[File:Elliko-skm.svg|250px|thumb|Approximation of an ellipse with osculating circles]] ...

...s://mathvault.ca/logarithm-theory/|title=The Ultimate Guide to Logarithm — Theory & Applications|date=8 May 2016|website=Math Vault|access-date=24 July 2019} ...letter {{mvar|e}} as the base of natural logarithms.<ref>{{citation|title=Theory of complex functions|last=Remmert, Reinhold.|date=1991|publisher=Springer-V ...

...nalytic number theory]] and has applications in [[physics]], [[probability theory]], and applied [[statistics]]. ...found by Euler, are [[rational number]]s and play an important role in the theory of [[modular form]]s. Many generalizations of the Riemann zeta function, su ...

In [[Sturm–Liouville theory|Sturm–Liouville form]] the differential equation is [[Vibronic coupling|Vibronic transitions]] in the Franck-Condon approximation can also be described using Laguerre polynomials.<ref>{{Cite journal|last=J ...