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Difference between revisions of "Inversion of an elliptic integral"

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The problem of constructing the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524601.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524602.png" /> or of constructing single-valued composite functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524603.png" /> in the case of an [[Elliptic integral|elliptic integral]]
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The problem of constructing the function $u$ as a function of $z$ or of constructing single-valued composite functions of the form $f(u(z))$ in the case of an [[Elliptic integral|elliptic integral]]
  
<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/i/i052/i052460/i0524604.png" /></td> </tr></table>
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$$z=\int\limits^uR(z,w)dz,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524605.png" /> is a rational function of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524607.png" /> which are related by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i0524609.png" /> is a polynomial of degree 3 or 4 without multiple roots. The complete solution of this problem was given almost simultaneously in 1827–1829 by N.H. Abel and C.G.J. Jacobi, who showed that its solution led to new transcendental elliptic functions (cf. [[Elliptic function|Elliptic function]]).
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where $R$ is a rational function of variables $z$ and $w$ which are related by the equation $w^2=F(z)$, where $F(z)$ is a polynomial of degree 3 or 4 without multiple roots. The complete solution of this problem was given almost simultaneously in 1827–1829 by N.H. Abel and C.G.J. Jacobi, who showed that its solution led to new transcendental elliptic functions (cf. [[Elliptic function|Elliptic function]]).
  
 
An essentially different approach to the theory of elliptic functions is due to K. Weierstrass. For the elliptic integral of the first kind in Weierstrass normal form,
 
An essentially different approach to the theory of elliptic functions is due to K. Weierstrass. For the elliptic integral of the first kind in Weierstrass normal 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/i/i052/i052460/i05246010.png" /></td> </tr></table>
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$$z=\int\limits^u\frac{dz}{w},\quad w^2=4z^3-g_2z-g_3,$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i05246011.png" /> turns out to be the Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i05246012.png" />-function with invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052460/i05246013.png" /> (see [[Weierstrass elliptic functions|Weierstrass elliptic functions]]). For the elliptic integral of the first kind in Legendre normal form,
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$u=\mathrm p(z)$ turns out to be the Weierstrass $\mathrm p$-function with invariants $g_2,g_3$ (see [[Weierstrass elliptic functions|Weierstrass elliptic functions]]). For the elliptic integral of the first kind in Legendre normal 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/i/i052/i052460/i05246014.png" /></td> </tr></table>
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$$z=\int\limits_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}},$$
  
 
inversion leads to the [[Jacobi elliptic functions|Jacobi elliptic functions]].
 
inversion leads to the [[Jacobi elliptic functions|Jacobi elliptic functions]].

Revision as of 12:35, 18 August 2014

The problem of constructing the function $u$ as a function of $z$ or of constructing single-valued composite functions of the form $f(u(z))$ in the case of an elliptic integral

$$z=\int\limits^uR(z,w)dz,$$

where $R$ is a rational function of variables $z$ and $w$ which are related by the equation $w^2=F(z)$, where $F(z)$ is a polynomial of degree 3 or 4 without multiple roots. The complete solution of this problem was given almost simultaneously in 1827–1829 by N.H. Abel and C.G.J. Jacobi, who showed that its solution led to new transcendental elliptic functions (cf. Elliptic function).

An essentially different approach to the theory of elliptic functions is due to K. Weierstrass. For the elliptic integral of the first kind in Weierstrass normal form,

$$z=\int\limits^u\frac{dz}{w},\quad w^2=4z^3-g_2z-g_3,$$

$u=\mathrm p(z)$ turns out to be the Weierstrass $\mathrm p$-function with invariants $g_2,g_3$ (see Weierstrass elliptic functions). For the elliptic integral of the first kind in Legendre normal form,

$$z=\int\limits_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}},$$

inversion leads to the Jacobi elliptic functions.

References

[1] N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)
[2] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2
How to Cite This Entry:
Inversion of an elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inversion_of_an_elliptic_integral&oldid=19005
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article