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One of the three fundamental [[Jacobi elliptic functions|Jacobi elliptic functions]]. It is denoted by
 
One of the three fundamental [[Jacobi elliptic functions|Jacobi elliptic functions]]. It is denoted by
  
<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/d0309401.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm dn} }  u  = { \mathop{\rm dn} } ( u , k )  = \Delta  { \mathop{\rm am} }  u .
 +
$$
  
 
The delta amplitude is expressed as follows in terms of the Weierstrass sigma-function, the Jacobi theta-functions or a series:
 
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>
+
$$
 +
{ \mathop{\rm dn} }  u  = { \mathop{\rm dn} } ( u , k )  = \
 +
 
 +
\frac{\sigma _ {2} ( u) }{\sigma _ {3} ( u) }
 +
  = \
  
<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>
+
\frac{\theta _ {0} ( 0) \theta _ {3} ( v) }{\theta _ {3} ( 0)
 +
\theta _ {0} ( v) }
 +
=
 +
$$
  
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,
+
$$
 +
= \
 +
1 - k  ^ {2}
 +
\frac{u  ^ {2} }{2!}
 +
+ k  ^ {2} (
 +
4 + k  ^ {2} )
 +
\frac{u  ^ {4} }{4!}
 +
- k  ^ {2} ( 16
 +
+ 44k  ^ {2} + k  ^ {4} )
 +
\frac{u  ^ {6} }{6!}
 +
- \dots ,
 +
$$
  
<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>
+
where  $  k $
 +
is the modulus of the delta amplitude,  $  0 \leq  k \leq  1 $,
 +
and  $  v = u /2 \omega $,
 +
$  2 \omega = \pi ( \theta _ {3} ( 0))  ^ {2} $.
 +
If  $  k= 0, 1 $
 +
one has, respectively,
 +
 
 +
$$
 +
{ \mathop{\rm dn} }  u  = 1 ,\ \
 +
{ \mathop{\rm dn} }  u  =
 +
\frac{1}{\cosh  u }
 +
.
 +
$$
  
 
See also [[Weierstrass elliptic functions|Weierstrass elliptic functions]]; [[Elliptic function|Elliptic function]].
 
See also [[Weierstrass elliptic functions|Weierstrass elliptic functions]]; [[Elliptic function|Elliptic function]].
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====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 17:32, 5 June 2020


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

$$ { \mathop{\rm dn} } u = { \mathop{\rm dn} } ( u , k ) = \Delta { \mathop{\rm am} } u . $$

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

$$ { \mathop{\rm dn} } u = { \mathop{\rm dn} } ( u , k ) = \ \frac{\sigma _ {2} ( u) }{\sigma _ {3} ( u) } = \ \frac{\theta _ {0} ( 0) \theta _ {3} ( v) }{\theta _ {3} ( 0) \theta _ {0} ( v) } = $$

$$ = \ 1 - k ^ {2} \frac{u ^ {2} }{2!} + k ^ {2} ( 4 + k ^ {2} ) \frac{u ^ {4} }{4!} - k ^ {2} ( 16 + 44k ^ {2} + k ^ {4} ) \frac{u ^ {6} }{6!} - \dots , $$

where $ k $ is the modulus of the delta amplitude, $ 0 \leq k \leq 1 $, and $ v = u /2 \omega $, $ 2 \omega = \pi ( \theta _ {3} ( 0)) ^ {2} $. If $ k= 0, 1 $ one has, respectively,

$$ { \mathop{\rm dn} } u = 1 ,\ \ { \mathop{\rm dn} } u = \frac{1}{\cosh u } . $$

See also Weierstrass elliptic functions; Elliptic function.

References

[1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) pp. Chapt. 3, Abschnitt 2

Comments

References

[a1] 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
How to Cite This Entry:
Delta amplitude. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delta_amplitude&oldid=46622
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article