View source for Delta amplitude

One of the three fundamental [[Jacobi elliptic functions|Jacobi elliptic functions]]. It is denoted by

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The delta amplitude is expressed as follows in terms of the Weierstrass sigma-function, the Jacobi theta-functions or a series:

<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/d/d030/d030940/d0309402.png" /></td> </tr></table>

<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/d/d030/d030940/d0309403.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030940/d0309404.png" /> is the modulus of the delta amplitude, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030940/d0309405.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030940/d0309406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030940/d0309407.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030940/d0309408.png" /> one has, respectively,

<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/d/d030/d030940/d0309409.png" /></td> </tr></table>

See also [[Weierstrass elliptic functions|Weierstrass elliptic functions]]; [[Elliptic function|Elliptic function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer  (1964)  pp. Chapt. 3, Abschnitt 2</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)  pp. Chapt. 13</TD></TR></table>