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{{Short description|Special function defined by an integral}} ...eonhard Euler]] ({{Circa|1750}}). Modern mathematics defines an "elliptic integral" as any [[function (mathematics)|function]] {{math|''f''}} which can be exp ...

{{Short description|Special function defined by an integral}} ...[[Fresnel diffraction]] phenomena and are defined through the following [[integral]] representations: ...

==Watson's contour integral== As an analogue of the [[Barnes integral]] for the hypergeometric series, [[G. N. Watson|Watson]] showed that ...

===Representation by an integral operator=== The [[integral transform]] ...

...egral]] of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of [[pi|{{pi}}]]). As a furth This Fourier integral, including the special case ...

...on ''α'' on the real numbers, we can define the [[Lebesgue–Stieltjes integral]] of a function ''f''. If this integral is finite for all polynomials ''f'', we can ...

...eries]]; nowadays the accepted and more general definition is via a [[line integral]] in the [[complex plane]], introduced in its full generality by [[Arthur E ...nal]] ''γ''/''δ'' equals just another G-function, and generalizations of [[integral transform]]s like the [[Hankel transform]] and the [[Laplace transform]] an ...

...ynomial interpolation on a segment using derivatives|Hermite interpolation|integral transform of Hermite polynomials|Hermite transform}} ...inatorics]], as an example of an [[Appell sequence]], obeying the [[umbral calculus]]; ...

The integral over the product of three associated Legendre polynomials (with orders matc The integral is zero unless ...

This integral is a [[special function|special]] (non-[[elementary function|elementary]]) The error function at +∞ is exactly 1 (see [[Gaussian integral]]). At the real axis, erf(''z'') approaches unity at ''z'' → +∞ a ...

...mbert ''W'' function from a result of Rosenlicht and of Liouville |journal=Integral Transforms and Special Functions |volume=19 |issue=10 |pages=709–712 |year= ==Calculus== ...

...sitive real part the gamma function is defined via a convergent [[improper integral]]: The gamma function then is defined as the [[analytic continuation]] of this integral function to a [[meromorphic function]] that is [[holomorphic function|holom ...

...uiv \sigma > 1,</math> the zeta function can be expressed by the following integral: <!-- This seemingly roundabout way of writing the integral makes it clear that the zeta function is a quotient of two Mellin transform ...

...use is widespread in mathematics and [[physics]], because of its simpler [[integral]] and [[derivative]]. The [[binary logarithm]] uses base {{math|2}} (that i ...3, 205.</ref> and the next year he connected it to the [[integral calculus|integral]] ...

...llipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae. ...\in[-a,a],</math> this curve is the top half of the ellipse. So twice the integral of <math>y(x)</math> over the interval <math>[-a,a]</math> will be the area ...