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A special case of elliptic functions (cf. [[Elliptic function|Elliptic function]]). They arise in the inversion of the elliptic integral of special form
 
A special case of elliptic functions (cf. [[Elliptic function|Elliptic function]]). They arise in the inversion of the elliptic integral of special form
  
<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/l/l058/l058120/l0581201.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { 0 } ^ { u }  ( 1 - t  ^ {4} )  ^ {-} 1/2  dt .
 +
$$
  
 
These integrals first appeared in the calculation of the arc length of the [[Bernoulli lemniscate|Bernoulli lemniscate]] in the work of G. Fagnano (1715). Lemniscate functions themselves were introduced by C.F. Gauss (1797).
 
These integrals first appeared in the calculation of the arc length of the [[Bernoulli lemniscate|Bernoulli lemniscate]] in the work of G. Fagnano (1715). Lemniscate functions themselves were introduced by C.F. Gauss (1797).
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There are two lemniscate functions:
 
There are two lemniscate functions:
  
<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/l/l058/l058120/l0581202.png" /></td> </tr></table>
+
$$
 +
= \cos  \mathop{\rm lemn}  z  =   \mathop{\rm cl}  z
 +
$$
  
 
and
 
and
  
<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/l/l058/l058120/l0581203.png" /></td> </tr></table>
+
$$
 +
\sin  \mathop{\rm lemn}  z  =   \mathop{\rm sl}  z  = \cos  \mathop{\rm lemn} \left (
 +
 
 +
\frac \omega {2}
 +
- z \right ) ,
 +
$$
  
 
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/l/l058/l058120/l0581204.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac \omega {2}
 +
  = \int\limits _ { 0 } ^ { 1 }  ( 1 - t  ^ {4} )  ^ {-} 1/2  dt  =
 +
\frac{
 +
\sqrt 2 }{8 \sqrt \pi }
 +
\left [ \Gamma \left (
 +
\frac{1}{4}
 +
\right )
 +
\right ]  ^ {2} .
 +
$$
  
The lemniscate functions can be expressed in terms of the [[Jacobi elliptic functions|Jacobi elliptic functions]] with modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058120/l0581205.png" />:
+
The lemniscate functions can be expressed in terms of the [[Jacobi elliptic functions|Jacobi elliptic functions]] with modulus $  k = \sqrt 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/l/l058/l058120/l0581206.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sl}  z  =
 +
\frac{\sqrt 2 }{2}
 +
 +
\frac{ \mathop{\rm sn}  ( z \sqrt 2 ) }{
 +
\mathop{\rm dn}  ( z \sqrt 2 ) }
 +
,\  \mathop{\rm cl}  z  =   \mathop{\rm cn}  ( z \sqrt 2 ) .
 +
$$
  
In the theory of [[Weierstrass elliptic functions|Weierstrass elliptic functions]] the lemniscate functions occur in the so-called harmonic case, when the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058120/l0581207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058120/l0581208.png" />.
+
In the theory of [[Weierstrass elliptic functions|Weierstrass elliptic functions]] the lemniscate functions occur in the so-called harmonic case, when the invariants $  g _ {2} = 4 $,  
 +
$  g _ {3} = 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR></table>

Revision as of 22:16, 5 June 2020


A special case of elliptic functions (cf. Elliptic function). They arise in the inversion of the elliptic integral of special form

$$ z = \int\limits _ { 0 } ^ { u } ( 1 - t ^ {4} ) ^ {-} 1/2 dt . $$

These integrals first appeared in the calculation of the arc length of the Bernoulli lemniscate in the work of G. Fagnano (1715). Lemniscate functions themselves were introduced by C.F. Gauss (1797).

There are two lemniscate functions:

$$ u = \cos \mathop{\rm lemn} z = \mathop{\rm cl} z $$

and

$$ \sin \mathop{\rm lemn} z = \mathop{\rm sl} z = \cos \mathop{\rm lemn} \left ( \frac \omega {2} - z \right ) , $$

where

$$ \frac \omega {2} = \int\limits _ { 0 } ^ { 1 } ( 1 - t ^ {4} ) ^ {-} 1/2 dt = \frac{ \sqrt 2 }{8 \sqrt \pi } \left [ \Gamma \left ( \frac{1}{4} \right ) \right ] ^ {2} . $$

The lemniscate functions can be expressed in terms of the Jacobi elliptic functions with modulus $ k = \sqrt 2 / 2 $:

$$ \mathop{\rm sl} z = \frac{\sqrt 2 }{2} \frac{ \mathop{\rm sn} ( z \sqrt 2 ) }{ \mathop{\rm dn} ( z \sqrt 2 ) } ,\ \mathop{\rm cl} z = \mathop{\rm cn} ( z \sqrt 2 ) . $$

In the theory of Weierstrass elliptic functions the lemniscate functions occur in the so-called harmonic case, when the invariants $ g _ {2} = 4 $, $ g _ {3} = 0 $.

References

[1] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2
How to Cite This Entry:
Lemniscate functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lemniscate_functions&oldid=17903
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article