Editing History of calculus (section)

===Integrals===
[[Niels Henrik Abel]] seems to have been the first to consider in a general way the question as to what [[differential equation]]s can be integrated in a finite form by the aid of ordinary functions, an investigation extended by [[Joseph Liouville|Liouville]]. [[Augustin Louis Cauchy|Cauchy]] early undertook the general theory of determining [[definite integral]]s, and the subject has been prominent during the 19th century. [[Frullani integral]]s, [[David Bierens de Haan]]'s work on the theory and his elaborate tables, [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]]'s lectures embodied in [[Friedrich Wilhelm Franz Meyer|Meyer]]'s treatise, and numerous memoirs of [[Adrien-Marie Legendre|Legendre]], [[Siméon Denis Poisson|Poisson]], [[Giovanni Antonio Amedeo Plana|Plana]], [[Joseph Ludwig Raabe|Raabe]], [[Leonhard Sohncke|Sohncke]], [[Oscar Xavier Schlömilch|Schlömilch]], [[Edwin Bailey Elliott|Elliott]], [[Charles Leudesdorf|Leudesdorf]] and [[Leopold Kronecker|Kronecker]] are among the noteworthy contributions.

[[Euler integral (disambiguation)|Eulerian integrals]] were first studied by [[Leonhard Euler|Euler]] and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

:<math>\int_0^1 x^{n-1}(1 - x)^{n-1} \, dx</math>

:<math>\int_0^\infty e^{-x} x^{n-1} \, dx</math>

although these were not the exact forms of Euler's study.

If ''n'' is a positive [[integer]]:
:<math>\int_0^\infty e^{-x}x^{n-1}dx = (n-1)!,</math>
but the integral converges for all positive real <math>n</math> and defines an [[analytic continuation]] of the [[factorial]] function to all of the [[complex plane]] except for poles at zero and the negative integers. To it Legendre assigned the symbol <math>\Gamma</math>, and it is now called the [[gamma function]]. Besides being analytic over positive reals <math>\mathbb{R}^{+}</math>, <math>\Gamma</math> also enjoys the uniquely defining property that <math>\log \Gamma</math> is [[Convex function|convex]], which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] has contributed an important theorem (Liouville, 1839), which has been elaborated by [[Joseph Liouville|Liouville]], [[Eugène Charles Catalan|Catalan]], [[Leslie Ellis]], and others. [[Joseph Ludwig Raabe|Raabe]] (1843–44), Bauer (1859), and [[Christoph Gudermann|Gudermann]] (1845) have written about the evaluation of <math>\Gamma (x)</math> and <math>\log \Gamma (x)</math>. Legendre's great table appeared in 1816.