: Mathématiques: elliptic integral

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mathematical analysis > calculus > integral calculus > elliptic integral

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mathematical physics > special function > hypergeometric function > elliptic integral

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elliptic integral

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.
Modern mathematics defines an "elliptic integral" as any function $f$ which can be expressed in the form
${\\displaystyle f(x)=\\int _{c}^{x}R{\\left({\ extstyle t,{\\sqrt {P(t)}}}\ ight)}\\,dt,}$
where $R$ is a rational function of its two arguments, $P$ is a polynomial of degree 3 or 4 with no repeated roots, and $c$ is a constant.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Elliptic_integral)

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