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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]] ...thematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | ...

...s one of the many [[zeta function]]s. It is formally defined for [[complex number|complex]] arguments ''s'' with Re(''s'') > 1 and ''q'' with Re('' ==Analytic continuation== ...

...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]]. ...

In [[digital signal processing]] and [[information theory]], the '''normalized sinc function''' is commonly defined for {{math|''x'' ...ood to be the limit value 1. The sinc function is then [[Analytic function|analytic]] everywhere and hence an [[entire function]]. ...

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 ...

...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]]. ...

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

...on''', which may then be defined over a wider domain of the argument by [[analytic continuation]]. The generalized hypergeometric series is sometimes just cal ...en the series defines an [[analytic function]]. Such a function, and its [[analytic continuation]]s, is called the '''hypergeometric function'''. ...

...ent algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic For complex arguments ''z'' with |''z''|&nbsp;≥&nbsp;1 it can be [[analytic continuation|analytically continued]] along any path in the complex plane t ...

...is one commonly used extension of the [[factorial function]] to [[complex number]]s. The gamma function is defined for all complex numbers except the non-po The gamma function then is defined as the [[analytic continuation]] of this integral function to a [[meromorphic function]] that ...

...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 == ...

...'' with |''z''| < 1; it can be extended to |''z''| ≥ 1 by the process of [[analytic continuation]]. The special case ''s'' = 1 involves the ordinary [[natural Going across the cut, if ''ε'' is an infinitesimally small positive real number, then: ...

...taken on the negative real axis and ''E''₁ can be defined by [[analytic continuation]] elsewhere on the complex plane. ...|200px|thumb| Relative error of the asymptotic approximation for different number <math>~N~</math> of terms in the truncated sum]] ...

...''s''.<ref>[http://dlmf.nist.gov/8.2.ii DLMF, Incomplete Gamma functions, analytic continuation]</ref> Complex analysis shows how properties of the real incom ...iemann surface]]. While this removes multi-valuedness, one has to know the theory behind it [http://math.berkeley.edu/~teleman/math/Riemann.pdf]; ...

...1782) are a system of complete and [[orthogonal polynomial]]s, with a vast number of mathematical properties, and numerous applications. They can be defined ...h>(-\infty,\infty)</math>, with weight functions that are the most natural analytic functions that ensure convergence of all integrals. ...

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

...hat number&nbsp;{{mvar|x}}. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since { ...e&nbsp;{{mvar|b}} is not equal to&nbsp;{{math|1}}, is always a unique real number&nbsp;{{mvar|y}}. More explicitly, the defining relation between exponentiat ...

...quation]]s. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. ...2017|url=https://books.google.com/books?id=yJ1YAAAAcAAJ&pg=PA421|series=4|number=279|doi=10.1080/14786447108640600}}</ref> ...

...se is measured by its [[eccentricity (mathematics)|eccentricity]] ''e'', a number ranging from ''e ='' 0 (the [[Limiting case (mathematics)|limiting case]] o [[Analytic geometry|Analytically]], the equation of a standard ellipse centered at the ...

"definition": "large number of other known integral", | <code>e</code> was interpreted as Euler's number ...