Namespaces
Variants
Actions

Difference between revisions of "Amplitude of an elliptic integral"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
−
The variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121601.png" />, considered as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121602.png" />, in an [[Elliptic integral|elliptic integral]] of the first kind
+
<!--
 +
a0121601.png
 +
$#A+1 = 11 n = 0
 +
$#C+1 = 11 : ~/encyclopedia/old_files/data/A012/A.0102160 Amplitude of an elliptic integral
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
−
<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/a/a012/a012160/a0121603.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
−
in the normal Legendre form. The concept of the amplitude of an elliptic integral and the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121604.png" /> were introduced by C.G.J. Jacobi in 1829. The amplitude of an elliptic integral is an infinite-valued periodic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121605.png" />. The basic elliptic Jacobi functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121608.png" /> are all single-valued. It is convenient, however (e.g. for purposes of tabulation), to consider an elliptic integral as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a0121609.png" /> of the amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a01216010.png" /> and the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012160/a01216011.png" />. See also [[Jacobi elliptic functions|Jacobi elliptic functions]].
+
The variable  $  \phi $,
 +
considered as a function of  $  z $,
 +
in an [[Elliptic integral|elliptic integral]] of the first kind
 +
 
 +
$$
 +
z  =  F ( \phi , k )  =  \int\limits _ { 0 } ^  \phi 
 +
 
 +
\frac{dt} \sqrt
 +
{1 - k  ^ {2}  \sin  ^ {2}  t }
 +
$$
 +
 
 +
in the normal Legendre form. The concept of the amplitude of an elliptic integral and the notation $  \phi = \mathop{\rm am}  z $
 +
were introduced by C.G.J. Jacobi in 1829. The amplitude of an elliptic integral is an infinite-valued periodic function of $  z $.  
 +
The basic elliptic Jacobi functions $  \sin  \mathop{\rm am}  z = \mathop{\rm sn}  z $,  
 +
$  \cos  \mathop{\rm am}  z = \mathop{\rm cn}  z $,  
 +
$  \Delta  \mathop{\rm am}  z = \mathop{\rm dn}  z $
 +
are all single-valued. It is convenient, however (e.g. for purposes of tabulation), to consider an elliptic integral as a function $  F ( \phi , k) $
 +
of the amplitude $  \phi $
 +
and the modulus $  k $.  
 +
See also [[Jacobi elliptic functions|Jacobi elliptic functions]].

Revision as of 16:10, 1 April 2020


The variable $ \phi $, considered as a function of $ z $, in an elliptic integral of the first kind

$$ z = F ( \phi , k ) = \int\limits _ { 0 } ^ \phi \frac{dt} \sqrt {1 - k ^ {2} \sin ^ {2} t } $$

in the normal Legendre form. The concept of the amplitude of an elliptic integral and the notation $ \phi = \mathop{\rm am} z $ were introduced by C.G.J. Jacobi in 1829. The amplitude of an elliptic integral is an infinite-valued periodic function of $ z $. The basic elliptic Jacobi functions $ \sin \mathop{\rm am} z = \mathop{\rm sn} z $, $ \cos \mathop{\rm am} z = \mathop{\rm cn} z $, $ \Delta \mathop{\rm am} z = \mathop{\rm dn} z $ are all single-valued. It is convenient, however (e.g. for purposes of tabulation), to consider an elliptic integral as a function $ F ( \phi , k) $ of the amplitude $ \phi $ and the modulus $ k $. See also Jacobi elliptic functions.

How to Cite This Entry:
Amplitude of an elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Amplitude_of_an_elliptic_integral&oldid=13707
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Amplitude_of_an_elliptic_integral&oldid=45096"