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Difference between revisions of "Chebyshev theorem on the integration of binomial differentials"

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The indefinite integral of the binomial differential
 
The indefinite integral of the binomial differential
 
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$$
<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/c/c022/c022000/c0220001.png" /></td> </tr></table>
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x^m (a+bx^n)^p
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220003.png" /> are real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220005.png" /> are rational numbers, cannot be expressed in terms of elementary functions for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220007.png" />, except in the case where (at least) one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022000/c0220009.png" /> is an integer. Obtained by P.L. Chebyshev (1853).
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where $a$ and $b$ are real numbers and $m$, $n$ and $p$ are rational numbers, cannot be expressed in terms of elementary functions for any $m$, $n$ and $p$, except in the case where (at least) one of $p$, $(m+1)/n$ and $p + (m+1)/n$ is an integer. Obtained by P.L. Chebyshev (1853).
  
  
  
 
====Comments====
 
====Comments====
See also [[Differential binomial|Differential binomial]].
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See also [[Differential binomial]].

Latest revision as of 21:00, 9 December 2014


The indefinite integral of the binomial differential $$ x^m (a+bx^n)^p $$ where $a$ and $b$ are real numbers and $m$, $n$ and $p$ are rational numbers, cannot be expressed in terms of elementary functions for any $m$, $n$ and $p$, except in the case where (at least) one of $p$, $(m+1)/n$ and $p + (m+1)/n$ is an integer. Obtained by P.L. Chebyshev (1853).


Comments

See also Differential binomial.

How to Cite This Entry:
Chebyshev theorem on the integration of binomial differentials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_theorem_on_the_integration_of_binomial_differentials&oldid=17835
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article