Elliptic functions
See [1][section 4.5] and [2] for more information.
1
Definition
An
elliptic function
is a singlevalued doublyperiodic function of a single complex variable
which is analytic except at poles and whose only singularities in the finite plane are poles.
Such functions are called elliptic because they define functions on the twotorus. Imagine
building a twotorus (a doughnut) by taking a square and identifying opposing sides – that
means a function on the complex plane which is periodic in two directions can be thought
of as a function on a square, periodic at opposing sides, and hence is a function on the
twotorus.
Given an angle
ϕ
, define
u
=
integraldisplay
ϕ
0
dθ
parenleftBig
1

m
sin
2
θ
parenrightBig
1
/
2
where
m
is the
parameter
, a real number in the interval 0
≤
m
≤
1. In terms of the elliptic
integrals discussed in AW section 5.8,
u
=
F
(sin
ϕ

m
) where
F
is the elliptic integral of the
first kind. The angle
ϕ
corresponding to
u
is known as the
amplitude
of
u
, and is denoted
am
u
. Then, define the
Jacobi elliptic functions
sn
u
=
sin
ϕ
cn
u
=
cos
ϕ
dn
u
=
parenleftBig
1

m
sin
2
ϕ
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 Spring '11
 smith
 elliptic functions, Elliptic integral, elliptic function, Jacobi's elliptic functions, Handbook of Mathematical Functions

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