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 .
Weierstrass sigma function
Plot of #Weierstrass sigma function using Domain coloring . The Weierstrass sigma function associated to a two-dimensional lattice
Λ
⊂
C
{\displaystyle \Lambda \subset \mathbb {C} }
σ
(
z
;
Λ
)
=
z
∏
w
∈
Λ
∗
(
1
−
z
w
)
e
z
/
w
+
1
2
(
z
/
w
)
2
{\displaystyle \sigma (z;\Lambda )=z\prod _{w\in \Lambda ^{*}}\left(1-{\frac {z}{w}}\right)e^{z/w+{\frac {1}{2}}(z/w)^{2}}}
where
Λ
∗
{\displaystyle \Lambda ^{*}}
Λ
−
{
0
}
{\displaystyle \Lambda -\{0\}}
fundamental pair of periods
Weierstrass zeta function
Plot of #Weierstrass 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 \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 \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}}
Eisenstein series of weight 2k + 2.
The derivative of the zeta function is
−
℘
(
z
)
{\displaystyle -\wp (z)}
℘
(
z
)
{\displaystyle \wp (z)}
Weierstrass elliptic function
The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory.
Weierstrass eta function
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} }
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 )}
w . The Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function .
Weierstrass p-function
Plot of #Weierstrass 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 \wp (z;\Lambda )=-\zeta '(z;\Lambda ),{\mbox{ for any }}z\in \mathbb {C} }
The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.
See also
This article incorporates material from Weierstrass sigma function on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
