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Elliptic functions (cf. [[Elliptic function|Elliptic function]]) resulting from the direct inversion of elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]) in Legendre normal form. This inversion problem was solved in 1827 independently by C.G.J. Jacobi and, in a slightly different form, by N.H. Abel. Jacobi's construction is based on an application of theta-functions (cf. [[Theta-function|Theta-function]]).

Let  $  \tau $
be a complex number with  $   \mathop{\rm Im}  \tau > 0 $. 
The Jacobi theta-functions  $  \theta _ {0} ( v) $, 
$  \theta _ {1} ( v) $, 
$  \theta _ {2} ( v) $, 
and  $  \theta _ {3} ( v) $
are represented by the following series in the complex variable  $  v $, 
which converge absolutely and uniformly on compact sets:

$$ 
\begin{array}
\theta _ {0} ( v)  = 
\theta _ {0} ( v;  \tau )  =& 
\sum _ {m = - \infty } ^  \infty  
(- 1)  ^ {m}
e ^ {i \pi m  ^ {2} \tau }
e ^ {2i \pi mv } \\
=&  1 + 2 \sum _ {m = 1 } ^  \infty   (- 1)  ^ {m} e ^ {i \pi m  ^ {2} \tau }  \cos ( 2 \pi mv); \\

\theta _ {1} ( v)  =  \theta _ {1} ( v;  \tau ) =&
i \sum _ {m = - \infty } ^  \infty   (- 1)  ^ {m} e ^ {i
\pi ( m - 1/2)  ^ {2} \tau } e ^ {( 2m - 1) i \pi v } &
= 2 \sum _ {m = 0 } ^  \infty   (- 1)  ^ {m} e ^ {i
\pi ( m + 1/2)  ^ {2} \tau }  \sin [( 2m + 1) \pi v]; \\

\theta _ {2} ( v)  =  \theta _ {2} ( v;  \tau )  =&
  \sum _ {m = - \infty } ^  \infty   e ^ {i \pi ( m - 1/2)  ^ {2} \tau } e ^ {( 2m - 1) i \pi v } \\
= 2 \sum _ {m = 0 } ^  \infty   e ^ {i \pi ( m + 1/2)  ^ {2} \tau }  \cos [( 2m + 1) \pi v]; \\

\theta _ {3} ( v)  =  \theta _ {3} ( v;  \tau )  =&
  \sum _ {m= - \infty } ^  \infty   e ^ {i \pi m  ^ {2} \tau } e ^ {2i \pi mv } \\
= 1 + 2 \sum _ {m = 1 } ^  \infty   e ^ {i \pi m  ^ {2} \tau }  \cos ( 2 \pi mv).
\end{array}
$$

These series converge fairly rapidly. The notation  $  \theta _ {0} ( v) $, 
$  \theta _ {1} ( v) $, 
$  \theta _ {2} ( v) $, 
$  \theta _ {3} ( v) $
goes back to r conditions','../w/w097460.htm','Weierstrass point','../w/w097490.htm','Weierstrass theorem','../w/w097510.htm','Weierstrass representation of a minimal surface','../w/w130040.htm')" style="background-color:yellow;">K. Weierstrass.  $  \theta _ {0} ( v) $
is often written  $  \theta _ {4} ( v) $, 
and there are other systems of notation. Jacobi used the notation  $  \Theta ( v) = \theta _ {0} ( v/2K) $, 
$  H ( v) = \theta _ {1} ( v/2K) $, 
$  H _ {1} ( v) = \theta _ {2} ( v/2K) $, 
and  $  \Theta _ {1} ( v) = \theta _ {3} ( v/2K) $, 
where  $  K = \pi \theta _ {3}  ^ {2} ( 0)/2 $.

All Jacobi theta-functions are entire transcendental functions of the complex variable  $  v $; 
$  \theta _ {1} ( v) $
is an odd function, and the other functions  $  \theta _ {0} ( v) $, 
$  \theta _ {2} ( v) $
and  $  \theta _ {3} ( v) $
are even.

The following periodicity relations hold:

$$ 
\theta _ {0} ( v \pm  1)  =  \theta _ {0} ( v),
$$

$$ 
\theta _ {0} ( v \pm  \tau )  =  - e ^ {i \pi \tau }
\cdot e ^ {\mps 2i \pi v } \cdot \theta _ {0} ( v);
$$

$$ 
\theta _ {1} ( v \pm  1)  =  - \theta _ {1} ( v),
$$

$$ 
\theta _ {1} ( v \pm  \tau )  =  - e ^ {- i \pi \tau
 } \cdot e ^ {\mps 2i \pi v } \cdot \theta _ {1} ( v);
$$

$$ 
\theta _ {2} ( v \pm  1)  =  - \theta _ {2} ( v),
$$

$$ 
\theta _ {2} ( v \pm  \tau )  =  e ^ {- i \pi \tau }
\cdot e ^ {\mps 2i \pi v } \cdot \theta _ {2} ( v);
$$

$$ 
\theta _ {3} ( v \pm  1)  =  \theta _ {3} ( v),
$$

$$ 
\theta _ {3} ( v \pm  \tau )  =  e ^ {- i \pi \tau }
\cdot e ^ {\mps 2i \pi v } \cdot \theta _ {3} ( v).
$$

These imply that the theta-functions are elliptic Hermite functions of the third kind (cf. also [[Hermite function|Hermite function]]).

The various theta-functions are connected by the following transformation formulas:

$$ 
\theta _ {0} \left ( v \pm  {
\frac{1}{2}
 } \right )  =  \theta _ {3} ( v),
$$

$$ 
\theta _ {0} \left ( v \pm  {
\frac{1}{2}
 } \tau \right )  = \ 
\pm  ie ^ {- i \pi \tau /4 }
\cdot e ^ {\mps i \pi v } \cdot \theta _ {1} ( v);
$$

$$ 
\theta _ {1} \left ( v \pm  {
\frac{1}{2}
 } \right )  =  \pm  \theta _ {2} ( v),
$$

$$ 
\theta _ {1} \left ( v \pm  {
\frac{1}{2}
 } \tau \right )  = \ 
\pm  ie ^ {- i \pi \tau /4 }
\cdot e ^ {\mps i \pi v } \cdot \theta _ {0} ( v);
$$

$$ 
\theta _ {2} \left ( v \pm  {
\frac{1}{2}
 } \right )  =  \theta _ {1} ( v),
$$

$$ 
\theta _ {2} \left ( v \pm  {
\frac{1}{2}
 } \tau \right )  =  e ^ 
{- i \pi \tau /4 } \cdot e ^ {\mps i \pi v } \cdot \theta _ {3} ( v);
$$

$$ 
\theta _ {3} \left ( v \pm  {
\frac{1}{2}
 } \right )  =  \theta _ {0} ( v),
$$

$$ 
\theta _ {3} \left ( v \pm  {
\frac{1}{2}
 } \tau \right )  =  e ^ 
{- i \pi \tau /4 } \cdot e ^ {\mps i \pi v } \cdot \theta _ {2} ( v).
$$

All four theta-functions satisfy one and the same differential equation (the [[Heat equation|heat equation]]):

$$ 

\frac{\partial   ^ {2} \theta }{\partial  v  ^ {2} }
  = \ 
4i \pi

\frac{\partial  \theta }{\partial  \tau }
 .
$$

The zero arguments of the theta-functions,  $  \theta _ {0} ( 0) $, 
$  \theta _ {1}  ^  \prime  ( 0) $, 
$  \theta _ {2} ( 0) $, 
$  \theta _ {3} ( 0) $
are important; here  $  \theta _ {1} ( 0) = 0 $. 
The relations between them are:

$$ 
\theta _ {1}  ^  \prime  ( 0)  = \ 
\pi \theta _ {0} ( 0) \theta _ {2} ( 0) \theta _ {3} ( 0),\ \ 
\theta _ {3}  ^ {4} ( 0)  = \ 
\theta _ {0}  ^ {4} ( 0) + \theta _ {2}  ^ {4} ( 0).
$$

In particular,

$$ 
\theta _ {0} ( 0)  =  H _ {0} H _ {3}  ^ {2} ,\ \ 
\theta _ {2} ( 0)  = \ 
2e ^ {i \pi \tau /4 }
H _ {0} H _ {1}  ^ {2} ,
$$

$$ 
\theta _ {3} ( 0)  =  H _ {0} H _ {2}  ^ {2} ,\  \theta _ {1}  ^  \prime  ( 0)  =  2 \pi e ^ {i \pi \tau /4 } H _ {0}  ^ {3} ,
$$

where

$$ 
H _ {0}  = \ 
\prod _ {m = 1 } ^  \infty  
( 1 - e ^ {2i \pi m \tau } ),
$$

$$ 
H _ {1}  =  \prod _ {m = 1 } ^  \infty   ( 1 + e ^ {2i \pi m \tau } ),
$$

$$ 
H _ {2}  =  \prod _ {m = 1 } ^  \infty   ( 1 + e ^ {( 2m - 1) i \pi \tau } ),
$$

$$ 
H _ {3}  =  \prod _ {m = 1 } ^  \infty   ( 1 - e ^ {( 2m - 1) i \pi \tau } ),
$$

$$ 
H _ {1} H _ {2} H _ {3}  =  1.
$$

The function  $  \theta _ {0} ( v) $
has simple zeros at  $  m + ( n - 1/2) \tau $; 
$  \theta _ {1} ( v) $
at  $  m + n \tau $; 
$  \theta _ {2} ( v) $
at  $  m - 1/2 + n \tau $; 
and  $  \theta _ {3} ( v) $
at  $  m - 1/2 + ( n - 1/2) \tau $; 
$  m, n = 0, \pm  1 , .  .  . $.

It is clear from the periodicity relations that certain ratios of theta-functions are elliptic functions in the proper sense. The basic Jacobi elliptic functions are:  $   \mathop{\rm sn}  u $([[
Sine amplitude|sine amplitude]]),  $   \mathop{\rm cn}  u $([[
Cosine amplitude|cosine amplitude]]) and  $   \mathop{\rm dn}  u $([[
Delta amplitude|delta amplitude]]). This notation was introduced by C. Gudermann (1838). The terminology stems from the old, and outdated, notation of Jacobi:  $   \mathop{\rm sn}  u = \sin   \mathop{\rm am}  u $, 
$   \mathop{\rm cn}  u = \cos   \mathop{\rm am}  u $, 
$   \mathop{\rm dn}  u = \Delta   \mathop{\rm am}  u $.

The new variable  $  u $
is connected with  $  v $
by  $  u = v \pi \theta _ {3}  ^ {2} ( 0) $. 
Denoting the modulus of the elliptic functions by  $  k = \theta _ {2}  ^ {2} ( 0)/ \theta _ {3}  ^ {2} ( 0) $, 
the Jacobi elliptic functions can be expressed in terms of theta-functions, or by means of power series that converge in a neighbourhood of the origin, as follows:

$$ 
 \mathop{\rm sn}  u  = \ 
 \mathop{\rm sn}  ( u;  k)  = \ 

\frac{\theta _ {3} ( 0) }{\theta _ {2} ( 0) }

\frac{\theta _ {1} ( v) }{\theta _ {0} ( v) }
 =
$$

$$ 
= \ 
u - ( 1 + k  ^ {2} ) 
\frac{u  ^ {3} }{3! }
 + ( 1 +
14k  ^ {2} + k  ^ {4} ) 
\frac{u  ^ {5} }{5! }
 - \dots ,
$$

$$ 
 \mathop{\rm cn}  u  =   \mathop{\rm cn}  ( u;  k)  =  
\frac{\theta _ {0} ( 0) }{\theta _ {2} ( 0) }
 
\frac{\theta _ {2} ( v) }{\theta _ {0} ( v) }
 =
$$

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

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

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

A convenient notation for inverses and ratios was introduced by J. Glaisher (1882):

$$ 
 \mathop{\rm ns}  u  =  
\frac{1}{ \mathop{\rm sn}  u }
 ,\ \ 
 \mathop{\rm nc}  u  =  
\frac{1}{ \mathop{\rm cn}  u }
 ,\ \ 
 \mathop{\rm nd}  u  =  
\frac{1}{ \mathop{\rm dn}  u }
 ,
$$

$$ 
 \mathop{\rm cs}  u  =  
\frac{ \mathop{\rm cn}  u }{ \mathop{\rm sn}  u }
 ,\ \ 
 \mathop{\rm ds}  u  =  
\frac{ \mathop{\rm dn}  u }{ \mathop{\rm sn}  u }
 ,\ \ 
 \mathop{\rm dc}  u  =  
\frac{ \mathop{\rm dn}  u }{ \mathop{\rm cn}  u }
 ,
$$

$$ 
 \mathop{\rm sc}  u  =  
\frac{ \mathop{\rm sn}  u }{ \mathop{\rm cn}  u }
 ,\ \ 
 \mathop{\rm sd}  u  =  
\frac{ \mathop{\rm sn}  u }{ \mathop{\rm dn}  u }
 ,\ \ 
 \mathop{\rm cd}  u  =  
\frac{ \mathop{\rm cn}  u }{ \mathop{\rm dn}  u }
 .
$$

The Jacobi elliptic functions  $   \mathop{\rm sn}  u $, 
$   \mathop{\rm cn}  u $
and  $   \mathop{\rm dn}  u $
are elliptic functions of the second order with the following periods:  $  4K $
and  $  2iK ^ { \prime } $
for  $   \mathop{\rm sn}  u $; 
$  4K $
and  $  2 ( K + iK ^ { \prime } ) $
for  $   \mathop{\rm cn}  u $; 
and  $  2K $
and  $  4iK ^ { \prime } $
for  $   \mathop{\rm dn}  u $, 
where  $  K = K ( k) = \pi \theta _ {3}  ^ {2} ( 0)/2 $
and  $  K ^ { \prime } = K ( k  ^  \prime  ) = - i \tau K $
are the values of the complete elliptic integrals of the first kind, and  $  k  ^  \prime  = ( 1 - k  ^ {2} )  ^ {1/2} $
is called the complementary modulus of the elliptic functions. The Jacobi elliptic functions have only simple poles, located at  $  2mK + ( 2n + 1) iK ^ { \prime } $; 
$  m, n = 0, \pm  1 , .  .  . $.

The three Jacobi elliptic functions are connected by the relations:

$$ 
 \mathop{\rm sn}  ^ {2} +  \mathop{\rm cn}  ^ {2}  u  =  1,
$$

$$ 
 \mathop{\rm dn}  ^ {2}  u + k  ^ {2}   \mathop{\rm sn}  ^ {2}  u  =  1 ,
$$

$$ 
 \mathop{\rm dn}  ^ {2}  u - k  ^ {2}   \mathop{\rm cn}  ^ {2}  u  =  1 - k  ^ {2}  =  k ^ {\prime 2 } ,
$$

$$ 
{
\frac{d}{du}
 }  \mathop{\rm sn}  u  =   \mathop{\rm cn}  u \cdot  \mathop{\rm dn}  u ,
$$

$$ 
{
\frac{d}{du}
 }  \mathop{\rm cn}  u  =  -  \mathop{\rm sn}  u \cdot  \mathop{\rm dn}  u ,
$$

$$ 
{
\frac{d}{du}
 }  \mathop{\rm dn}  u  =  - k  ^ {2}   \mathop{\rm sn}  u \cdot  \mathop{\rm cn}  u ;
$$

and by the differential equations:

$$ 
\left ( {
\frac{d}{du}
 }  \mathop{\rm sn}  u \right )  ^ {2}  = \ 
( 1 -  \mathop{\rm sn}  ^ {2}  u) ( 1 - k  ^ {2}   \mathop{\rm sn}  ^ {2}  u),
$$

$$ 
\left ( {
\frac{d}{du}
 }  \mathop{\rm cn}  u \right )  ^ {2}  =  ( 1 -
 \mathop{\rm cn}  ^ {2}  u) ( k ^ {\prime 2 } + k  ^ {2}   \mathop{\rm cn}  ^ {2}  u),
$$

$$ 
\left ( {
\frac{d}{du}
 }  \mathop{\rm dn}  u \right )  ^ {2}  =  ( 1 -
 \mathop{\rm dn}  ^ {2}  u) (  \mathop{\rm dn}  ^ {2}  u - k ^ {\prime 2 } ).
$$

The addition theorems for Jacobi elliptic functions are:

$$ 
 \mathop{\rm sn} ( u _ {1} + u _ {2} )  = \ 

\frac{ \mathop{\rm sn}  u _ {1} \cdot  \mathop{\rm cn}  u _ {2} \cdot
 \mathop{\rm dn}  u _ {2} +  \mathop{\rm sn}  u _ {2} \cdot
 \mathop{\rm cn}  u _ {1} \cdot  \mathop{\rm dn}  u _ {1} }{1 - k  ^ {2}   \mathop{\rm sn}  ^ {2}  u _ {1} \cdot  \mathop{\rm sn}  ^ {2}  u _ {2} }
 ,
$$

$$ 
 \mathop{\rm cn} ( u _ {1} + u _ {2} )  =  
\frac{ \mathop{\rm cn}  u _ {1} \cdot
 \mathop{\rm cn}  u _ {2} -  \mathop{\rm sn}  u _ {1} \cdot  \mathop{\rm dn}  u _ {1} \cdot  \mathop{\rm sn}  u _ {2} \cdot  \mathop{\rm dn}  u _ {2} }{1 - k
 ^ {2}   \mathop{\rm sn}  ^ {2}  u _ {1} \cdot  \mathop{\rm sn}  ^ {2}  u _ {2} }
 ,
$$

$$ 
 \mathop{\rm dn} ( u _ {1} + u _ {2} )  =  
\frac{ \mathop{\rm dn}  u _ {1} \cdot  \mathop{\rm dn} \ 
u _ {2} - k  ^ {2}   \mathop{\rm sn}  u _ {1} \cdot  \mathop{\rm cn} \ 
u _ {1} \cdot  \mathop{\rm sn}  u _ {2} \cdot  \mathop{\rm cn}  u _ {2} }{1 - k
 ^ {2}   \mathop{\rm sn}  ^ {2}  u _ {1} \cdot  \mathop{\rm sn}  ^ {2}  u _ {2} }
 .
$$

The connection between Jacobi elliptic functions and elliptic integrals is as follows. If

$$ 
u  =  \int\limits _ { 0 } ^ { z } 

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

\frac{ds }{\sqrt {1 - k  ^ {2}  \sin  ^ {2}  s } }

$$

is an elliptic integral of the first kind in Legendre normal form, then its inverse is  $  z =  \mathop{\rm sn}  u $; 
this constitutes the starting point of the Jacobi theory. The variable  $  \phi =  \mathop{\rm am}  u $
is an infinitely-valued function of  $  u $, 
and is called the amplitude of the elliptic integral  $  u $,

$$ 
 \mathop{\rm cn}  u  = \ 
\sqrt {1 -  \mathop{\rm sn}  ^ {2}  u }  =  \cos   \mathop{\rm am}  u ,
$$

$$ 
 \mathop{\rm dn}  u  =  \sqrt {1 - k  ^ {2}   \mathop{\rm sn}  ^ {2} \ 
u }  =  \sqrt {1 - k  ^ {2}  \sin  ^ {2}   \mathop{\rm am}  u } .
$$

The main relations between the constants are:

$$ 
K  =  K ( k)  =  {
\frac{1}{2}
 }
\pi \theta _ {3}  ^ {2} ( 0;  \tau ),\ \ 
K ^ { \prime }  =  K ( k  ^  \prime  ),
$$

$$ 
\tau K  =  iK ^ { \prime } ,\  k  =  
\frac{\theta _ {2}  ^ {2} ( 0) }{\theta _ {3}  ^ {2} ( 0) }
 ,\  k  ^  \prime 
 =  
\frac{\theta _ {0}  ^ {2} ( 0) }{\theta _ {3}  ^ {2} ( 0) }
 .
$$

In applied problems the modulus  $  k $
is usually known and very often the normal case  $  0 < k < 1 $
holds, or the complementary modulus  $  k  ^  \prime  = ( 1 - k  ^ {2} )  ^ {1/2} $, 
$  0 < k < 1 $, 
is given. It is required to find  $  K $, 
$  K ^ { \prime } $, 
and  $  \tau $
or  $  q = e ^ {i \pi \tau } $. 
By setting  $  2 \epsilon = ( 1 - \sqrt {k  ^  \prime  } )/( 1 + \sqrt {k  ^  \prime  } ) $, 
where  $  0 < k < 1 $, 
one has  $  0 < \epsilon < 1/2 $. 
To find  $  q $
the following series, which converges rapidly if  $  0 < \epsilon < 1 / 2 $, 
is used:

$$ 
q  =  e ^ {i \pi \tau }  = \ 
\epsilon + 2 \epsilon  ^ {5} + 15 \epsilon  ^ {9} + 150 \epsilon  ^ {13} +
O ( \epsilon  ^ {17} ).
$$

The numbers  $  K $
and  $  K ^ { \prime } $
of the complete elliptic integrals are determined by the formulas

$$ 
K  =  {
\frac{1}{2}
 }
\pi \theta _ {3}  ^ {2} ( 0;  \tau )  = \ 
{
\frac \pi {2}
 } \left \{
1 + 2 \sum _ {m = 1 } ^  \infty  
q ^ {m  ^ {2} } \right \} ,
$$

$$ 
K ^ { \prime }  =  {
\frac{1} \pi 
 } K   \mathop{\rm ln}  {
\frac{1}{q}
 } ,
$$

or are found from tables.

When  $  k = 0 $
or  $  k = 1 $
the Jacobi elliptic functions degenerate into trigonometric or hyperbolic functions, respectively:

$$ 
 \mathop{\rm sn} ( u;  0)  =  \sin  u ,\ \ 
 \mathop{\rm cn} ( u;  0)  =  \cos  u ,
$$

$$ 
 \mathop{\rm dn}  ( u;  0)  =  1; \   \mathop{\rm sn}  ( u;  1)  =   \mathop{\rm tanh}  u ,
$$

$$ 
 \mathop{\rm cn}  ( u;  1)  =   \mathop{\rm dn}  ( u;  1)  =  
\frac{1}{\cosh  u }
 .
$$

Theoretically, a simpler construction of the theory of elliptic functions was given by Weierstrass (1862–1863) (see [[Weierstrass elliptic functions|Weierstrass elliptic functions]]). For a given  $  k $, 
$  0 < k < 1 $, 
the Weierstrass invariants  $  e _ {1} , e _ {2} , e _ {3} $
are found from, for example, the formulas:  $  e _ {1} = ( 2 - k  ^ {2} )/3 $, 
$  e _ {2} = ( 2k  ^ {2} - 1)/3 $, 
$  e _ {3} = - ( 1 + k  ^ {2} )/3 $, 
and then  $  g _ {2} = - 4 ( e _ {1} e _ {2} + e _ {2} e _ {3} + e _ {3} e _ {1} ) $
and  $  g _ {3} = 4e _ {1} e _ {2} e _ {3} $
can be found. The half-periods are defined by  $  \omega _ {1} = K/( e _ {1} - e _ {3} )  ^ {1/2} $
and  $  \omega _ {3} = iK ^ { \prime } /( e _ {1} - e _ {3} )  ^ {1/2} $; 
these enable one to calculate all the remaining quantities related to the Weierstrass elliptic functions.

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  C.G.J. Jacobi,   "Fundamenta nova theoriae functionum ellipticarum" , Königsberg  (1829)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  C.G.J. Jacobi,   "Fundamenta nova theoriae functionum ellipticarum" , ''Gesammelte Werke'' , '''1''' , Chelsea, reprint  (1969)  pp. 49–239</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,   "Elements of the theory of elliptic functions" , Amer. Math. Soc.  (1990)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer  (1964)  pp. Chapt.8</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Enneper,   "Elliptische Funktionen. Theorie und Geschichte" , Halle  (1890)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Tannéry,   J. Molk,   "Eléments de la théorie des fonctions elliptiques" , '''1–2''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.M. Zhuravskii,   "Handbook on elliptic functions" , Moscow-Leningrad  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   F. Lösch,   "Tafeln höheren Funktionen" , Teubner  (1966)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Mumford,   "Tata lectures on Theta" , '''1–2''' , Birkhäuser  (1983–1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Weil,   "Elliptic functions according to Eisenstein and Kronecker" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Bowman,   "Introduction to elliptic functions with applications" , Dover, reprint  (1961)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.-i. Igusa,   "Theta functions" , Springer  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Hancock,   "Theory of elliptic functions" , Dover, reprint  (1958)</TD></TR></table>