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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]]).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540501.png" /> be a complex number with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540502.png" />. The Jacobi theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540505.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540506.png" /> are represented by the following series in the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j0540507.png" />, which converge absolutely and uniformly on compact sets:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/j/j054/j054050/j0540508.png" /></td> </tr></table>
+
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]]).
  
<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/j/j054/j054050/j0540509.png" /></td> </tr></table>
+
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:
  
<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/j/j054/j054050/j05405010.png" /></td> </tr></table>
+
$$
 +
\begin{eqnarray*}
 +
\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); \\
  
<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/j/j054/j054050/j05405011.png" /></td> </tr></table>
+
\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]; \\
  
<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/j/j054/j054050/j05405012.png" /></td> </tr></table>
+
\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]; \\
  
<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/j/j054/j054050/j05405013.png" /></td> </tr></table>
+
\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{eqnarray*}
 +
$$
  
<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/j/j054/j054050/j05405014.png" /></td> </tr></table>
+
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 $.
  
<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/j/j054/j054050/j05405015.png" /></td> </tr></table>
+
All Jacobi theta-functions are entire transcendental functions of the complex variable $  v $;  
 
+
$  \theta _ {1} ( v) $
<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/j/j054/j054050/j05405016.png" /></td> </tr></table>
+
is an odd function, and the other functions $  \theta _ {0} ( v) $,  
 
+
$  \theta _ {2} ( v) $
These series converge fairly rapidly. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405020.png" /> 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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405021.png" /> is often written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405022.png" />, and there are other systems of notation. Jacobi used the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405027.png" />.
+
and $  \theta _ {3} ( v) $
 
+
are even.
All Jacobi theta-functions are entire transcendental functions of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405028.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405029.png" /> is an odd function, and the other functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405032.png" /> are even.
 
  
 
The following periodicity relations hold:
 
The following periodicity relations hold:
  
<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/j/j054/j054050/j05405033.png" /></td> </tr></table>
+
$$
 +
\begin{eqnarray*}
 +
\theta _ {0} ( v \pm  1)  &=&  \theta _ {0} ( v), \\
  
<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/j/j054/j054050/j05405034.png" /></td> </tr></table>
+
\theta _ {0} ( v \pm  \tau ) &=- e ^ {i \pi \tau }
 +
\cdot e ^ {\mp 2i \pi v } \cdot \theta _ {0} ( v); \\
  
<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/j/j054/j054050/j05405035.png" /></td> </tr></table>
+
\theta _ {1} ( v \pm  1) &=- \theta _ {1} ( v), \\
  
<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/j/j054/j054050/j05405036.png" /></td> </tr></table>
+
\theta _ {1} ( v \pm  \tau )  &=&  - e ^ {- i \pi \tau
 +
} \cdot e ^ {\mp 2i \pi v } \cdot \theta _ {1} ( v); \\
  
<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/j/j054/j054050/j05405037.png" /></td> </tr></table>
+
\theta _ {2} ( v \pm  1)  &=- \theta _ {2} ( v), \\
  
<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/j/j054/j054050/j05405038.png" /></td> </tr></table>
+
\theta _ {2} ( v \pm  \tau )  &=&  e ^ {- i \pi \tau }
 +
\cdot e ^ {\mp 2i \pi v } \cdot \theta _ {2} ( v); \\
  
<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/j/j054/j054050/j05405039.png" /></td> </tr></table>
+
\theta _ {3} ( v \pm  1)  &=&  \theta _ {3} ( v), \\
  
<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/j/j054/j054050/j05405040.png" /></td> </tr></table>
+
\theta _ {3} ( v \pm  \tau )  &=&  e ^ {- i \pi \tau }
 +
\cdot e ^ {\mp 2i \pi v } \cdot \theta _ {3} ( v).
 +
\end{eqnarray*}
 +
$$
  
 
These imply that the theta-functions are elliptic Hermite functions of the third kind (cf. also [[Hermite function|Hermite function]]).
 
These imply that the theta-functions are elliptic Hermite functions of the third kind (cf. also [[Hermite function|Hermite function]]).
Line 47: Line 97:
 
The various theta-functions are connected by the following transformation formulas:
 
The various theta-functions are connected by the following transformation formulas:
  
<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/j/j054/j054050/j05405041.png" /></td> </tr></table>
+
$$
 +
\theta _ {0} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \right )  = \theta _ {3} ( v),
 +
$$
  
<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/j/j054/j054050/j05405042.png" /></td> </tr></table>
+
$$
 +
\theta _ {0} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \tau \right )  = \
 +
\pm  ie ^ {- i \pi \tau /4 }
 +
\cdot e ^ {\mps i \pi v } \cdot \theta _ {1} ( v);
 +
$$
  
<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/j/j054/j054050/j05405043.png" /></td> </tr></table>
+
$$
 +
\theta _ {1} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \right )  = \pm  \theta _ {2} ( v),
 +
$$
  
<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/j/j054/j054050/j05405044.png" /></td> </tr></table>
+
$$
 +
\theta _ {1} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \tau \right )  = \
 +
\pm  ie ^ {- i \pi \tau /4 }
 +
\cdot e ^ {\mps i \pi v } \cdot \theta _ {0} ( v);
 +
$$
  
<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/j/j054/j054050/j05405045.png" /></td> </tr></table>
+
$$
 +
\theta _ {2} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \right )  = \theta _ {1} ( v),
 +
$$
  
<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/j/j054/j054050/j05405046.png" /></td> </tr></table>
+
$$
 +
\theta _ {2} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \tau \right )  = e ^
 +
{- i \pi \tau /4 } \cdot e ^ {\mps i \pi v } \cdot \theta _ {3} ( v);
 +
$$
  
<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/j/j054/j054050/j05405047.png" /></td> </tr></table>
+
$$
 +
\theta _ {3} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \right )  = \theta _ {0} ( v),
 +
$$
  
<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/j/j054/j054050/j05405048.png" /></td> </tr></table>
+
$$
 +
\theta _ {3} \left ( v \pm  {
 +
\frac{1}{2}
 +
} \tau \right )  = e ^
 +
{- i \pi \tau /4 } \cdot e ^ {\mps i \pi v } \cdot \theta _ {2} ( v).
 +
$$
  
 
All four theta-functions satisfy one and the same differential equation (the [[Heat equation|heat equation]]):
 
All four theta-functions satisfy one and the same differential equation (the [[Heat equation|heat equation]]):
  
<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/j/j054/j054050/j05405049.png" /></td> </tr></table>
+
$$
  
The zero arguments of the theta-functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405053.png" /> are important; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405054.png" />. The relations between them are:
+
\frac{\partial  ^ {2} \theta }{\partial  v  ^ {2} }
 +
  = \
 +
4i \pi
  
<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/j/j054/j054050/j05405055.png" /></td> </tr></table>
+
\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,
 
In particular,
  
<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/j/j054/j054050/j05405056.png" /></td> </tr></table>
+
$$
 +
\theta _ {0} ( 0)  = H _ {0} H _ {3}  ^ {2} ,\ \
 +
\theta _ {2} ( 0= \
 +
2e ^ {i \pi \tau /4 }
 +
H _ {0} H _ {1}  ^ {2} ,
 +
$$
  
<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/j/j054/j054050/j05405057.png" /></td> </tr></table>
+
$$
 +
\theta _ {3} ( 0)  = H _ {0} H _ {2}  ^ {2} ,\  \theta _ {1}  ^  \prime  ( 0= 2 \pi e ^ {i \pi \tau /4 } H _ {0}  ^ {3} ,
 +
$$
  
 
where
 
where
  
<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/j/j054/j054050/j05405058.png" /></td> </tr></table>
+
$$
 
+
H _ {0= \
<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/j/j054/j054050/j05405059.png" /></td> </tr></table>
+
\prod _ {m = 1 } ^  \infty 
 +
( 1 - e ^ {2i \pi m \tau } ),
 +
$$
  
<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/j/j054/j054050/j05405060.png" /></td> </tr></table>
+
$$
 +
H _ {1}  = \prod _ {m = 1 } ^  \infty  ( 1 + e ^ {2i \pi m \tau } ),
 +
$$
  
<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/j/j054/j054050/j05405061.png" /></td> </tr></table>
+
$$
 +
H _ {2}  = \prod _ {m = 1 } ^  \infty  ( 1 + e ^ {( 2m - 1) i \pi \tau } ),
 +
$$
  
<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/j/j054/j054050/j05405062.png" /></td> </tr></table>
+
$$
 +
H _ {3}  = \prod _ {m = 1 } ^  \infty  ( 1 - e ^ {( 2m - 1) i \pi \tau } ),
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405063.png" /> has simple zeros at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405064.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405065.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405066.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405067.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405068.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405069.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405070.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405071.png" />.
+
$$
 +
H _ {1} H _ {2} H _ {3}  = 1.
 +
$$
  
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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405072.png" /> ([[Sine amplitude|sine amplitude]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405073.png" /> ([[Cosine amplitude|cosine amplitude]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405074.png" /> ([[Delta amplitude|delta amplitude]]). This notation was introduced by C. Gudermann (1838). The terminology stems from the old, and outdated, notation of Jacobi: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405077.png" />.
+
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 , \dots$.
  
The new variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405078.png" /> is connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405079.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405080.png" />. Denoting the modulus of the elliptic functions by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405081.png" />, 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:
+
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 $.
  
<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/j/j054/j054050/j05405082.png" /></td> </tr></table>
+
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:
  
<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/j/j054/j054050/j05405083.png" /></td> </tr></table>
+
$$
 +
\begin{eqnarray*}
 +
\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 , \\
  
<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/j/j054/j054050/j05405084.png" /></td> </tr></table>
+
\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 ,\\\
  
<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/j/j054/j054050/j05405085.png" /></td> </tr></table>
+
\mathop{\rm dn}  u  =   \mathop{\rm dn}  ( u; k)  &=
 
+
\frac{\theta _ {0} ( 0) }{\theta _ {3} ( 0) } \frac{\theta _ {3} ( v) }{\theta _ {0} ( v) } \\
<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/j/j054/j054050/j05405086.png" /></td> </tr></table>
+
&=& 1 - k  ^ {2}
 
+
\frac{u  ^ {2} }{2! }
<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/j/j054/j054050/j05405087.png" /></td> </tr></table>
+
+ k  ^ {2} ( 4 + k  ^ {2}
 +
)
 +
\frac{u  ^ {4} }{4! }
 +
- k  ^ {2} ( 16 + 44k  ^ {2} + k  ^ {4} )
 +
\frac{u  ^ {6} }{6! }
 +
+ \dots .
 +
\end{eqnarray*}
 +
$$
  
 
A convenient notation for inverses and ratios was introduced by J. Glaisher (1882):
 
A convenient notation for inverses and ratios was introduced by J. Glaisher (1882):
  
<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/j/j054/j054050/j05405088.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ns}  u  =
 +
\frac{1}{ \mathop{\rm sn}  u }
 +
,\ \
 +
\mathop{\rm nc}  u  =
 +
\frac{1}{ \mathop{\rm cn}  u }
 +
,\ \
 +
\mathop{\rm nd}  u  =
 +
\frac{1}{ \mathop{\rm dn}  u }
 +
,
 +
$$
  
<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/j/j054/j054050/j05405089.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cs}  u  =
 +
\frac{ \mathop{\rm cn}  u }{ \mathop{\rm sn}  u }
 +
,\ \
 +
\mathop{\rm ds}  u  =
 +
\frac{ \mathop{\rm dn}  u }{ \mathop{\rm sn}  u }
 +
,\ \
 +
\mathop{\rm dc}  u  =
 +
\frac{ \mathop{\rm dn}  u }{ \mathop{\rm cn}  u }
 +
,
 +
$$
  
<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/j/j054/j054050/j05405090.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sc}  u  =
 +
\frac{ \mathop{\rm sn}  u }{ \mathop{\rm cn}  u }
 +
,\ \
 +
\mathop{\rm sd}  u  =
 +
\frac{ \mathop{\rm sn}  u }{ \mathop{\rm dn}  u }
 +
,\ \
 +
\mathop{\rm cd}  u  =
 +
\frac{ \mathop{\rm cn}  u }{ \mathop{\rm dn}  u }
 +
.
 +
$$
  
The Jacobi elliptic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405093.png" /> are elliptic functions of the second order with the following periods: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405095.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405096.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j05405099.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050101.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050104.png" /> are the values of the complete elliptic integrals of the first kind, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050105.png" /> is called the complementary modulus of the elliptic functions. The Jacobi elliptic functions have only simple poles, located at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050106.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050107.png" />.
+
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:
 
The three Jacobi elliptic functions are connected by the relations:
  
<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/j/j054/j054050/j054050108.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sn}  ^ {2} +  \mathop{\rm cn}  ^ {2}  u  = 1,
 +
$$
  
<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/j/j054/j054050/j054050109.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dn}  ^ {2}  u + k  ^ {2}  \mathop{\rm sn}  ^ {2}  u  = 1 ,
 +
$$
  
<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/j/j054/j054050/j054050110.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dn}  ^ {2}  u - k  ^ {2}  \mathop{\rm cn}  ^ {2}  u  = 1 - k  ^ {2}  = k ^ {\prime 2 } ,
 +
$$
  
<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/j/j054/j054050/j054050111.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{d}{du}
 +
}  \mathop{\rm sn}  u  =   \mathop{\rm cn}  u \cdot  \mathop{\rm dn}  u ,
 +
$$
  
<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/j/j054/j054050/j054050112.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{d}{du}
 +
}  \mathop{\rm cn}  u  = - \mathop{\rm sn}  u \cdot  \mathop{\rm dn}  u ,
 +
$$
  
<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/j/j054/j054050/j054050113.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{d}{du}
 +
}  \mathop{\rm dn}  u  = - k  ^ {2}  \mathop{\rm sn}  u \cdot  \mathop{\rm cn}  u ;
 +
$$
  
 
and by the differential equations:
 
and by the differential equations:
  
<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/j/j054/j054050/j054050114.png" /></td> </tr></table>
+
$$
 +
\left ( {
 +
\frac{d}{du}
 +
}  \mathop{\rm sn}  u \right )  ^ {2}  = \
 +
( 1 -  \mathop{\rm sn}  ^ {2}  u) ( 1 - k  ^ {2}  \mathop{\rm sn}  ^ {2}  u),
 +
$$
  
<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/j/j054/j054050/j054050115.png" /></td> </tr></table>
+
$$
 +
\left ( {
 +
\frac{d}{du}
 +
}  \mathop{\rm cn}  u \right )  ^ {2}  = ( 1 -
 +
\mathop{\rm cn}  ^ {2}  u) ( k ^ {\prime 2 } + k  ^ {2}  \mathop{\rm cn}  ^ {2}  u),
 +
$$
  
<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/j/j054/j054050/j054050116.png" /></td> </tr></table>
+
$$
 +
\left ( {
 +
\frac{d}{du}
 +
}  \mathop{\rm dn}  u \right )  ^ {2}  = ( 1 -
 +
\mathop{\rm dn}  ^ {2}  u) (  \mathop{\rm dn}  ^ {2}  u - k ^ {\prime 2 } ).
 +
$$
  
 
The addition theorems for Jacobi elliptic functions are:
 
The addition theorems for Jacobi elliptic functions are:
  
<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/j/j054/j054050/j054050117.png" /></td> </tr></table>
+
$$
 +
\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} }
 +
,
 +
$$
  
<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/j/j054/j054050/j054050118.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cn} ( u _ {1} + u _ {2} )  =
 +
\frac{ \mathop{\rm cn}  u _ {1} \cdot
 +
\mathop{\rm cn}  u _ {2} -  \mathop{\rm sn}  u _ {1} \cdot  \mathop{\rm dn}  u _ {1} \cdot  \mathop{\rm sn}  u _ {2} \cdot  \mathop{\rm dn}  u _ {2} }{1 - k
 +
^ {2}  \mathop{\rm sn}  ^ {2}  u _ {1} \cdot  \mathop{\rm sn}  ^ {2}  u _ {2} }
 +
,
 +
$$
  
<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/j/j054/j054050/j054050119.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dn} ( u _ {1} + u _ {2} )  =
 +
\frac{ \mathop{\rm dn}  u _ {1} \cdot  \mathop{\rm dn} \
 +
u _ {2} - k  ^ {2}  \mathop{\rm sn}  u _ {1} \cdot  \mathop{\rm cn} \
 +
u _ {1} \cdot  \mathop{\rm sn}  u _ {2} \cdot  \mathop{\rm cn}  u _ {2} }{1 - k
 +
^ {2}  \mathop{\rm sn}  ^ {2}  u _ {1} \cdot  \mathop{\rm sn}  ^ {2}  u _ {2} }
 +
.
 +
$$
  
 
The connection between Jacobi elliptic functions and elliptic integrals is as follows. If
 
The connection between Jacobi elliptic functions and elliptic integrals is as follows. If
  
<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/j/j054/j054050/j054050120.png" /></td> </tr></table>
+
$$
 +
= \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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050121.png" />; this constitutes the starting point of the Jacobi theory. The variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050122.png" /> is an infinitely-valued function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050123.png" />, and is called the amplitude of the elliptic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050124.png" />,
+
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 $,
  
<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/j/j054/j054050/j054050125.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cn}  u  = \
 +
\sqrt {1 - \mathop{\rm sn}  ^ {2}  u }  = \cos  \mathop{\rm am}  u ,
 +
$$
  
<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/j/j054/j054050/j054050126.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dn}  u  = \sqrt {1 - k  ^ {2}  \mathop{\rm sn}  ^ {2} \
 +
u }  = \sqrt {1 - k  ^ {2}  \sin  ^ {2}  \mathop{\rm am}  u } .
 +
$$
  
 
The main relations between the constants are:
 
The main relations between the constants are:
  
<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/j/j054/j054050/j054050127.png" /></td> </tr></table>
+
$$
 +
= K ( k)  = {
 +
\frac{1}{2}
 +
}
 +
\pi \theta _ {3}  ^ {2} ( 0; \tau ),\ \
 +
K ^ { \prime }  = K ( k  ^  \prime  ),
 +
$$
  
<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/j/j054/j054050/j054050128.png" /></td> </tr></table>
+
$$
 +
\tau K  = iK ^ { \prime } ,\  k  =
 +
\frac{\theta _ {2}  ^ {2} ( 0) }{\theta _ {3}  ^ {2} ( 0) }
 +
,\  k  ^  \prime
 +
=
 +
\frac{\theta _ {0}  ^ {2} ( 0) }{\theta _ {3}  ^ {2} ( 0) }
 +
.
 +
$$
  
In applied problems the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050129.png" /> is usually known and very often the normal case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050130.png" /> holds, or the complementary modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050132.png" />, is given. It is required to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050134.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050135.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050136.png" />. By setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050137.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050138.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050139.png" />. To find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050140.png" /> the following series, which converges rapidly if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050141.png" />, is used:
+
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:
  
<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/j/j054/j054050/j054050142.png" /></td> </tr></table>
+
$$
 +
= e ^ {i \pi \tau }  = \
 +
\epsilon + 2 \epsilon  ^ {5} + 15 \epsilon  ^ {9} + 150 \epsilon  ^ {13} +
 +
O ( \epsilon  ^ {17} ).
 +
$$
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050144.png" /> of the complete elliptic integrals are determined by the formulas
+
The numbers $  K $
 +
and $  K ^ { \prime } $
 +
of the complete elliptic integrals are determined by the formulas
  
<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/j/j054/j054050/j054050145.png" /></td> </tr></table>
+
$$
 +
= {
 +
\frac{1}{2}
 +
}
 +
\pi \theta _ {3}  ^ {2} ( 0; \tau )  = \
 +
{
 +
\frac \pi {2}
 +
} \left \{
 +
1 + 2 \sum _ {m = 1 } ^  \infty 
 +
q ^ {m  ^ {2} } \right \} ,
 +
$$
  
<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/j/j054/j054050/j054050146.png" /></td> </tr></table>
+
$$
 +
K ^ { \prime }  = {
 +
\frac{1} \pi
 +
} K  \mathop{\rm ln}  {
 +
\frac{1}{q}
 +
} ,
 +
$$
  
 
or are found from tables.
 
or are found from tables.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050147.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050148.png" /> the Jacobi elliptic functions degenerate into trigonometric or hyperbolic functions, respectively:
+
When $  k = 0 $
 +
or $  k = 1 $
 +
the Jacobi elliptic functions degenerate into trigonometric or hyperbolic functions, 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/j/j054/j054050/j054050149.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sn} ( u; 0)  = \sin  u ,\ \
 +
\mathop{\rm cn} ( u; 0= \cos  u ,
 +
$$
  
<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/j/j054/j054050/j054050150.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dn}  ( u; 0)  = 1; \  \mathop{\rm sn}  ( u; 1)  =   \mathop{\rm tanh}  u ,
 +
$$
  
<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/j/j054/j054050/j054050151.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm cn}  ( u; 1)  =   \mathop{\rm dn}  ( u; 1)  =
 +
\frac{1}{\cosh  u }
 +
.
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050153.png" />, the Weierstrass invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050154.png" /> are found from, for example, the formulas: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050157.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050159.png" /> can be found. The half-periods are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050160.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054050/j054050161.png" />; these enable one to calculate all the remaining quantities related to the Weierstrass elliptic functions.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 02:00, 20 June 2022


Elliptic functions (cf. Elliptic function) resulting from the direct inversion of elliptic integrals (cf. 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).

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:

$$ \begin{eqnarray*} \theta _ {0} ( v) = \theta _ {0} ( v; \tau ) &=& \sum _ {m = - \infty } ^ \infty (- 1) ^ {m} e ^ {i \pi m ^ {2} \tau } e ^ {2i \pi mv } \\ &=& 1 + 2 \sum _ {m = 1 } ^ \infty (- 1) ^ {m} e ^ {i \pi m ^ {2} \tau } \cos ( 2 \pi mv); \\ \theta _ {1} ( v) = \theta _ {1} ( v; \tau ) &=& i \sum _ {m = - \infty } ^ \infty (- 1) ^ {m} e ^ {i \pi ( m - 1/2) ^ {2} \tau } e ^ {( 2m - 1) i \pi v } \\ &=& 2 \sum _ {m = 0 } ^ \infty (- 1) ^ {m} e ^ {i \pi ( m + 1/2) ^ {2} \tau } \sin [( 2m + 1) \pi v]; \\ \theta _ {2} ( v) = \theta _ {2} ( v; \tau ) &=& \sum _ {m = - \infty } ^ \infty e ^ {i \pi ( m - 1/2) ^ {2} \tau } e ^ {( 2m - 1) i \pi v } \\ &=& 2 \sum _ {m = 0 } ^ \infty e ^ {i \pi ( m + 1/2) ^ {2} \tau } \cos [( 2m + 1) \pi v]; \\ \theta _ {3} ( v) = \theta _ {3} ( v; \tau ) &=& \sum _ {m= - \infty } ^ \infty e ^ {i \pi m ^ {2} \tau } e ^ {2i \pi mv } \\ &=& 1 + 2 \sum _ {m = 1 } ^ \infty e ^ {i \pi m ^ {2} \tau } \cos ( 2 \pi mv). \end{eqnarray*} $$

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:

$$ \begin{eqnarray*} \theta _ {0} ( v \pm 1) &=& \theta _ {0} ( v), \\ \theta _ {0} ( v \pm \tau ) &=& - e ^ {i \pi \tau } \cdot e ^ {\mp 2i \pi v } \cdot \theta _ {0} ( v); \\ \theta _ {1} ( v \pm 1) &=& - \theta _ {1} ( v), \\ \theta _ {1} ( v \pm \tau ) &=& - e ^ {- i \pi \tau } \cdot e ^ {\mp 2i \pi v } \cdot \theta _ {1} ( v); \\ \theta _ {2} ( v \pm 1) &=& - \theta _ {2} ( v), \\ \theta _ {2} ( v \pm \tau ) &=& e ^ {- i \pi \tau } \cdot e ^ {\mp 2i \pi v } \cdot \theta _ {2} ( v); \\ \theta _ {3} ( v \pm 1) &=& \theta _ {3} ( v), \\ \theta _ {3} ( v \pm \tau ) &=& e ^ {- i \pi \tau } \cdot e ^ {\mp 2i \pi v } \cdot \theta _ {3} ( v). \end{eqnarray*} $$

These imply that the theta-functions are elliptic Hermite functions of the third kind (cf. also 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):

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

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 , \dots$.

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), $ \mathop{\rm cn} u $(cosine amplitude) and $ \mathop{\rm dn} u $(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:

$$ \begin{eqnarray*} \mathop{\rm sn} u = \mathop{\rm sn} ( u; k) &=& \frac{\theta _ {3} ( 0) }{\theta _ {2} ( 0) } \frac{\theta _ {1} ( v) }{\theta _ {0} ( v) } \\ &=& u - ( 1 + k ^ {2} ) \frac{u ^ {3} }{3! } + ( 1 + 14k ^ {2} + k ^ {4} ) \frac{u ^ {5} }{5! } - \dots , \\ \mathop{\rm cn} u = \mathop{\rm cn} ( u; k) &=& \frac{\theta _ {0} ( 0) }{\theta _ {2} ( 0) } \frac{\theta _ {2} ( v) }{\theta _ {0} ( v) } \\ &=& 1 - \frac{u ^ {2} }{2! } + ( 1 + 4k ^ {2} ) \frac{u ^ {4} }{4! } - ( 1 + 44k ^ {2} + 16k ^ {4} ) \frac{u ^ {6} }{6! } + \dots ,\\\ \mathop{\rm dn} u = \mathop{\rm dn} ( u; k) &=& \frac{\theta _ {0} ( 0) }{\theta _ {3} ( 0) } \frac{\theta _ {3} ( v) }{\theta _ {0} ( v) } \\ &=& 1 - k ^ {2} \frac{u ^ {2} }{2! } + k ^ {2} ( 4 + k ^ {2} ) \frac{u ^ {4} }{4! } - k ^ {2} ( 16 + 44k ^ {2} + k ^ {4} ) \frac{u ^ {6} }{6! } + \dots . \end{eqnarray*} $$

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). 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

[1a] C.G.J. Jacobi, "Fundamenta nova theoriae functionum ellipticarum" , Königsberg (1829)
[1b] C.G.J. Jacobi, "Fundamenta nova theoriae functionum ellipticarum" , Gesammelte Werke , 1 , Chelsea, reprint (1969) pp. 49–239
[2] N.I. Akhiezer, "Elements of the theory of elliptic functions" , Amer. Math. Soc. (1990) (Translated from Russian)
[3] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8
[4] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[5] A. Enneper, "Elliptische Funktionen. Theorie und Geschichte" , Halle (1890)
[6] J. Tannéry, J. Molk, "Eléments de la théorie des fonctions elliptiques" , 1–2 , Chelsea, reprint (1972)
[7] A.M. Zhuravskii, "Handbook on elliptic functions" , Moscow-Leningrad (1941) (In Russian)
[8] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)

Comments

References

[a1] D. Mumford, "Tata lectures on Theta" , 1–2 , Birkhäuser (1983–1984)
[a2] A. Weil, "Elliptic functions according to Eisenstein and Kronecker" , Springer (1976)
[a3] F. Bowman, "Introduction to elliptic functions with applications" , Dover, reprint (1961)
[a4] J.-i. Igusa, "Theta functions" , Springer (1972)
[a5] H. Hancock, "Theory of elliptic functions" , Dover, reprint (1958)
How to Cite This Entry:
Jacobi elliptic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_elliptic_functions&oldid=11940
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article