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...pproximation]] to the [[prime-counting function]], which is defined as the number of [[prime numbers]] less than or equal to a given value <math>x</math>. [[ ...ic integral has an integral representation defined for all positive [[real number]]s {{mvar|x}}&nbsp;≠&nbsp;1 by the [[integral|definite integral]] ...

|journal=Quart. J. Math. | year=1966 | volume=17 | number=1 | pages=7–10 |doi=10.1088/0305-4470/18/10/014 |volume=18 | number=10 | pages=1583 ...

...thematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | where the product is over all [[prime number]]s.<ref>{{harvnb|Apostol|1976|loc=Theorem 11.7}}</ref> ...

...after [[mathematician]] [[Ernest William Barnes]].<ref>E. W. Barnes, "The theory of the G-function", ''Quarterly Journ. Pure and Appl. Math.'' '''31''' (190 ...N. Watson]], "[[A Course of Modern Analysis]]", CUP.</ref> the [[Bernoulli number]] <math>B_{2k}</math> would have been written as <math>(-1)^{k+1} B_k </mat ...

...s one of the many [[zeta function]]s. It is formally defined for [[complex number|complex]] arguments ''s'' with Re(''s'')&nbsp;>&nbsp;1 and ''q'' with Re('' | journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory ...

and if <math> p</math> is a number with <math>0 < p < \pi </math>, the equation <math> f(t+p) = f(t) </math> m ...precise form of the solutions to Hill's equation is described by [[Floquet theory]]. Solutions can also be written in terms of Hill determinants. ...

...the ordinary [[derivative]] operator). For the Bernoulli polynomials, the number of crossings of the ''x''-axis in the [[unit interval]] does not go up with for ''n'' ≥ 0, where ''B''_''k'' are the [[Bernoulli number]]s, and ''E''_''k'' are the [[Euler numbers]]. ...

...elements are independent of the total angular momentum projection quantum number (<math>m_9</math>): |title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers ...

...t1=Tyurin|first1=Andrey N.|title=Quantization, Classical and Quantum Field Theory and Theta-Functions|eprint=math/0210466v1|date=30 October 2002}}</ref> ...theory this comes from a [[line bundle]] condition of [[descent (category theory)|descent]]. ...

In [[digital signal processing]] and [[information theory]], the '''normalized sinc function''' is commonly defined for {{math|''x'' ...e book |first=Phillip M. |last=Woodward |title=Probability and information theory, with applications to radar |url=https://archive.org/details/probabilityinf ...

for [[complex number]] inputs {{math|''x'', ''y''}} such that {{math|Re ''x'' > 0, Re ''y'' > 0} where in the last identity {{mvar|n}} is any positive real number. (One may move from the first integral to the second one by substituting <m ...

...ll polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of [[orthogonality]] in the usual *[[Sturm–Liouville theory|Sturm-Liouville theory]] ...

The complex number ''q'' is called the '''accessory parameter'''. Heun's equation has four [[r *{{Citation | last1=Forsyth | first1=Andrew Russell | title=Theory of differential equations. 4. Ordinary linear equations | origyear=1906 | p ...

...hysics (in particular, the theory of [[random matrices]]), [[approximation theory]], [[numerical analysis]], and many others. The relations above define <math>P_n</math> up to multiplication by a number. Various normalisations are used to fix the constant, e.g. ...

...tion in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For the ...nd Legendre functions with applications to integral equations of potential theory | url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 | publisher= ...

...nalytic number theory]] and has applications in [[physics]], [[probability theory]], and applied [[statistics]]. ...stablished a relation between its [[Root of a function|zeros]] and [[prime number theorem|the distribution of prime numbers]].<ref>This paper also contained ...

...|200px|thumb| Relative error of the asymptotic approximation for different number <math>~N~</math> of terms in the truncated sum]] ...otted on the figure to the right for various values of <math>N</math>, the number of terms in the truncated sum (<math>N=1</math> in red, <math>N=5</math> in ...

...movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane. The functions with a double pole at ''z ...described in detail by {{harvs|last=Boutroux|year1=1913|year2=1914}}. The number of poles in a ball of radius ''R'' grows roughly like a constant times ''R' ...

In [[integral calculus]], an '''elliptic integral''' is one of a number of related functions defined as the value of certain integrals. Originally ...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 ...

...given (real) <math>q</math> such periodic solutions exist for an infinite number of values of <math>a</math>, called ''characteristic numbers'', conventiona == Floquet theory == ...