Mathematical functions related to Weierstrass's elliptic function
In mathematics , the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function . They are named for Karl Weierstrass . The relation between the sigma, zeta, and
℘
{\displaystyle \wp }
functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.
Weierstrass sigma function [ edit ]
Plot of the sigma function using Domain coloring .
The Weierstrass sigma function associated to a two-dimensional lattice
Λ
⊂
C
{\displaystyle \Lambda \subset \mathbb {C} }
is defined to be the product
σ
(
z
;
Λ
)
=
z
∏
w
∈
Λ
∗
(
1
−
z
w
)
exp
(
z
w
+
1
2
(
z
w
)
2
)
=
z
∏
m
,
n
=
−
∞
{
m
,
n
}
≠
0
∞
(
1
−
z
m
ω
1
+
n
ω
2
)
exp
(
z
m
ω
1
+
n
ω
2
+
1
2
(
z
m
ω
1
+
n
ω
2
)
2
)
{\displaystyle {\begin{aligned}\operatorname {\sigma } {(z;\Lambda )}&=z\prod _{w\in \Lambda ^{*}}\left(1-{\frac {z}{w}}\right)\exp \left({\frac {z}{w}}+{\frac {1}{2}}\left({\frac {z}{w}}\right)^{2}\right)\\[5mu]&=z\prod _{\begin{smallmatrix}m,n=-\infty \\\{m,n\}\neq 0\end{smallmatrix}}^{\infty }\left(1-{\frac {z}{m\omega _{1}+n\omega _{2}}}\right)\exp {\left({\frac {z}{m\omega _{1}+n\omega _{2}}}+{\frac {1}{2}}\left({\frac {z}{m\omega _{1}+n\omega _{2}}}\right)^{2}\right)}\end{aligned}}}
where
Λ
∗
{\displaystyle \Lambda ^{*}}
denotes
Λ
−
{
0
}
{\displaystyle \Lambda -\{0\}}
oder
{
ω
1
,
ω
2
}
{\displaystyle \{\omega _{1},\omega _{2}\}}
are a fundamental pair of periods .
Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is
σ
(
z
;
Λ
)
=
ω
i
π
exp
(
η
i
z
2
ω
i
)
sin
(
π
z
ω
i
)
∏
n
=
1
∞
(
1
−
sin
2
(
π
z
/
ω
i
)
sin
2
(
n
π
ω
j
/
ω
i
)
)
{\displaystyle \operatorname {\sigma } {(z;\Lambda )}={\frac {\omega _{i}}{\pi }}\exp {\left({\frac {\eta _{i}z^{2}}{\omega _{i}}}\right)}\sin {\left({\frac {\pi z}{\omega _{i}}}\right)}\prod _{n=1}^{\infty }\left(1-{\frac {\sin ^{2}{\left(\pi z/\omega _{i}\right)}}{\sin ^{2}{\left(n\pi \omega _{j}/\omega _{i}\right)}}}\right)}
for any
{
i
,
j
}
∈
{
1
,
2
,
3
}
{\displaystyle \{i,j\}\in \{1,2,3\}}
with
i
≠
j
{\displaystyle i\neq j}
and where we have used the notation
η
i
=
ζ
(
ω
i
/
2
;
Λ
)
{\displaystyle \eta _{i}=\zeta (\omega _{i}/2;\Lambda )}
(see zeta function below).
Weierstrass zeta function [ edit ]
Plot of the zeta function using Domain coloring
The Weierstrass zeta function is defined by the sum
ζ
(
z
;
Λ
)
=
σ
′
(
z
;
Λ
)
σ
(
z
;
Λ
)
=
1
z
+
∑
w
∈
Λ
∗
(
1
z
−
w
+
1
w
+
z
w
2
)
.
{\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {\sigma '(z;\Lambda )}{\sigma (z;\Lambda )}}={\frac {1}{z}}+\sum _{w\in \Lambda ^{*}}\left({\frac {1}{z-w}}+{\frac {1}{w}}+{\frac {z}{w^{2}}}\right).}
The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as:
ζ
(
z
;
Λ
)
=
1
z
−
∑
k
=
1
∞
G
2
k
+
2
(
Λ
)
z
2
k
+
1
{\displaystyle \operatorname {\zeta } {(z;\Lambda )}={\frac {1}{z}}-\sum _{k=1}^{\infty }{\mathcal {G}}_{2k+2}(\Lambda )z^{2k+1}}
where
G
2
k
+
2
{\displaystyle {\mathcal {G}}_{2k+2}}
is the Eisenstein series of weight 2k + 2.
The derivative of the zeta function is
−
℘
(
z
)
{\displaystyle -\wp (z)}
, where
℘
(
z
)
{\displaystyle \wp (z)}
is the Weierstrass elliptic function .
The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory.
Weierstrass eta function [ edit ]
The Weierstrass eta function is defined to be
η
(
w
;
Λ
)
=
ζ
(
z
+
w
;
Λ
)
−
ζ
(
z
;
Λ
)
,
for any
z
∈
C
{\displaystyle \eta (w;\Lambda )=\zeta (z+w;\Lambda )-\zeta (z;\Lambda ),{\mbox{ for any }}z\in \mathbb {C} }
and any w in the lattice
Λ
{\displaystyle \Lambda }
This is well-defined, i.e.
ζ
(
z
+
w
;
Λ
)
−
ζ
(
z
;
Λ
)
{\displaystyle \zeta (z+w;\Lambda )-\zeta (z;\Lambda )}
only depends on the lattice vector w . The Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function .
Weierstrass ℘-function[ edit ]
Plot of the p-function using Domain coloring
The Weierstrass p-function is related to the zeta function by
℘
(
z
;
Λ
)
=
−
ζ
′
(
z
;
Λ
)
,
for any
z
∈
C
{\displaystyle \operatorname {\wp } {(z;\Lambda )}=-\operatorname {\zeta '} {(z;\Lambda )},{\mbox{ for any }}z\in \mathbb {C} }
The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.
Consider the situation where one period is real, which we can scale to be
ω
1
=
2
π
{\displaystyle \omega _{1}=2\pi }
and the other is taken to the limit of
ω
2
→
i
∞
{\displaystyle \omega _{2}\rightarrow i\infty }
so that the functions are only singly-periodic. The corresponding invariants are
{
g
2
,
g
3
}
=
{
1
12
,
1
216
}
{\displaystyle \{g_{2},g_{3}\}=\left\{{\tfrac {1}{12}},{\tfrac {1}{216}}\right\}}
of discriminant
Δ
=
0
{\displaystyle \Delta =0}
. Then we have
η
1
=
π
12
{\displaystyle \eta _{1}={\tfrac {\pi }{12}}}
and thus from the above infinite product definition the following equality:
σ
(
z
;
Λ
)
=
2
e
z
2
/
24
sin
(
z
2
)
{\displaystyle \operatorname {\sigma } {(z;\Lambda )}=2e^{z^{2}/24}\sin {\left({\tfrac {z}{2}}\right)}}
A generalization for other sine-like functions on other doubly-periodic lattices is
f
(
z
)
=
π
ω
1
e
−
(
4
η
1
/
ω
1
)
z
2
σ
(
2
z
;
Λ
)
{\displaystyle f(z)={\frac {\pi }{\omega _{1}}}e^{-(4\eta _{1}/\omega _{1})z^{2}}\operatorname {\sigma } {(2z;\Lambda )}}
This article incorporates material from Weierstrass sigma function on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .