Namespaces
Variants
Actions

Difference between revisions of "Cosine amplitude"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
c0266401.png
 +
$#A+1 = 11 n = 0
 +
$#C+1 = 11 : ~/encyclopedia/old_files/data/C026/C.0206640 Cosine amplitude,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''elliptic cosine''
 
''elliptic cosine''
  
 
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], denoted by
 
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], 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/c/c026/c026640/c0266401.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cn}  u  = \
 +
\mathop{\rm cn} ( u , k)  = \
 +
\cosam  u .
 +
$$
  
 
The cosine amplitude is expressible in terms of the Weierstrass sigma-functions, the Jacobi theta-functions or a power series, as follows:
 
The cosine amplitude is expressible in terms of the Weierstrass sigma-functions, the Jacobi theta-functions or a power series, as follows:
  
<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/c026/c026640/c0266402.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cn}  u  = \
 +
\mathop{\rm cn} ( u, k)  = \
 +
 
 +
\frac{\sigma _ {1} ( u) }{\sigma _ {3} ( u) }
 +
  = \
 +
 
 +
\frac{\theta _ {0} ( 0) \theta _ {2} ( \upsilon ) }{\theta _ {2} ( 0) \theta _ {0} ( \upsilon ) }
 +
=
 +
$$
  
<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/c026/c026640/c0266403.png" /></td> </tr></table>
+
$$
 +
= \
 +
1 -
 +
\frac{u  ^ {2} }{2! }
 +
+ ( 1 + 4k  ^ {2} )
 +
\frac{u
 +
^ {4} }{4! }
 +
- ( 1 + 44k  ^ {2} + 16k  ^ {4} )
 +
\frac{u  ^ {6} }{6! }
 +
+ \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266404.png" /> is the modulus of the elliptic function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266405.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266406.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266407.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266408.png" /> one has, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c0266409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c02664010.png" />.
+
where $  k $
 +
is the modulus of the elliptic function, 0 \leq  k \leq  1 $;  
 +
$  \upsilon = u/2 \omega $,  
 +
and $  2 \omega = \pi \theta _ {3}  ^ {2} ( 0) $.  
 +
For $  k = 0, 1 $
 +
one has, respectively, $  \mathop{\rm cn} ( u , 0) = \cos  u $,
 +
$  \mathop{\rm cn} ( u , 1) = 1/ \cosh  u $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''2''' , Springer  (1964)  pp. Chapt. 3</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''2''' , Springer  (1964)  pp. Chapt. 3</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
More on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026640/c02664011.png" />, e.g. derivatives, evenness, behaviour on the real line, etc. can be found in [[#References|[a1]]].
+
More on the function $  \mathop{\rm cn}  u $,  
 +
e.g. derivatives, evenness, behaviour on the real line, etc. can be found in [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


elliptic cosine

One of the three basic Jacobi elliptic functions, denoted by

$$ \mathop{\rm cn} u = \ \mathop{\rm cn} ( u , k) = \ \cosam u . $$

The cosine amplitude is expressible in terms of the Weierstrass sigma-functions, the Jacobi theta-functions or a power series, as follows:

$$ \mathop{\rm cn} u = \ \mathop{\rm cn} ( u, k) = \ \frac{\sigma _ {1} ( u) }{\sigma _ {3} ( u) } = \ \frac{\theta _ {0} ( 0) \theta _ {2} ( \upsilon ) }{\theta _ {2} ( 0) \theta _ {0} ( \upsilon ) } = $$

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

where $ k $ is the modulus of the elliptic function, $ 0 \leq k \leq 1 $; $ \upsilon = u/2 \omega $, and $ 2 \omega = \pi \theta _ {3} ^ {2} ( 0) $. For $ k = 0, 1 $ one has, respectively, $ \mathop{\rm cn} ( u , 0) = \cos u $, $ \mathop{\rm cn} ( u , 1) = 1/ \cosh u $.

References

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

Comments

More on the function $ \mathop{\rm cn} u $, e.g. derivatives, evenness, behaviour on the real line, etc. can be found in [a1].

References

[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Cosine amplitude. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cosine_amplitude&oldid=46531
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article