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Difference between revisions of "Pseudo-elliptic integral"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756701.png" /> be a rational function of two variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756702.png" /> a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form
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{{TEX|done}}
 
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Let $R(\cdot ,\cdot)$ be a rational function of two variables and $f(z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756703.png" /></td> </tr></table>
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\begin{equation*}
 
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\int R(z,\sqrt{f(z)})\,dz,
which can be expressed elementarily, that is, by algebraic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756704.png" /> or in the logarithms of such functions. For example,
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\end{equation*}
 
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which can be expressed elementarily, that is, by algebraic functions in $z$ or in the logarithms of such functions. For example,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756705.png" /></td> </tr></table>
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\begin{equation*}
 
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\int \frac{z^3\,dz}{\sqrt{z^4-1}}
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\end{equation*}
 
is a pseudo-elliptic integral. See [[Elliptic integral|Elliptic integral]].
 
is a pseudo-elliptic integral. See [[Elliptic integral|Elliptic integral]].

Latest revision as of 09:09, 7 December 2012

Let $R(\cdot ,\cdot)$ be a rational function of two variables and $f(z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form \begin{equation*} \int R(z,\sqrt{f(z)})\,dz, \end{equation*} which can be expressed elementarily, that is, by algebraic functions in $z$ or in the logarithms of such functions. For example, \begin{equation*} \int \frac{z^3\,dz}{\sqrt{z^4-1}} \end{equation*} is a pseudo-elliptic integral. See Elliptic integral.

How to Cite This Entry:
Pseudo-elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-elliptic_integral&oldid=16948
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article