View source for Cosine amplitude

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''elliptic cosine''

One of the three basic [[Jacobi elliptic functions|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====
<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====
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====
<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>