Difference between revisions of "Pseudo-elliptic integral"
From Encyclopedia of Mathematics
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| − | Let | + | {{TEX|done}} |
| − | + | 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, | |
| − | which can be expressed elementarily, that is, by algebraic functions in | + | \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|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
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