Weierstrass functions explained

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

\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

Λ \subset\Complex is defined to be the product\begin{align} \operatorname{\sigma}{(z; Λ )} & = z\prod _{w\in Λ ^{*}} \left( 1- (z)/(w) \right) \exp\left( (zw)/(+) (12 \left( (zw\right))/(^{2\right)}))/(\\[ 5 mu] & =) z\prod _{\begin{smallmatrix} m,n =-inf ty \ \{m,n\} ≠ 0 \end{smallmatrix}} ^{infty \left(1} - (z)/(\right) \exp{\left( m\omega _{1} + n\omega _{2}) (z)/(m\omega _{1} + n\omega _{2}) + (1)/(\left( 2) (z)/(m\omega _{1} + n\omega _{2}) \right) ^{2\right)} \end{align}} where Λ ^{*} denotes Λ- \{ 0 \} or \{\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

\operatorname{\sigma}{(z; Λ )} = (\omega _{i})/(\exp{\left( \pi) (η _{i} z ^{2})/(\omega _{i}) \right)} \sin{\left( (\pi z)/(\omega _{i}) \right)}\prod ^{inf ty\left( 1- (\sin ^{2{\left(\pi} z/\omega _{i\right)})/()} _{n =1} }\right)for any \{i,j\}\in\{ 1 , 2 , 3 \} with i ≠ j and where we have used the notation η _{i=\zeta(\omega} _{i/2;Λ)} (see zeta function below).

Weierstrass zeta function

The Weierstrass zeta function is defined by the sum

\operatorname{\zeta}{(z; Λ )} = (\sigma'(z; Λ ))/(= \sigma(z; Λ )) (1)/(z) + \sum _{w\in Λ ^{*}} \left( (1)/(+ z - w) (1)/(+ w) (z)/(w ^{2}) \right).

The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as:

\operatorname{\zeta}{(z; Λ )} = (1)/(z) - \sum ^{inf ty} _{k =1} l{G} _{2k+2} ( Λ )z ^{2k+1} where l{G} _{2k+2} is the Eisenstein series of weight 2k + 2.

The derivative of the zeta function is

- \wp(z) , where \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

The Weierstrass eta function is defined to be

η (w; Λ ) = \zeta(z + w; Λ ) - \zeta(z; Λ ), forany z \in \Complex and any w in the lattice Λ

This is well-defined, i.e.

\zeta(z + w; Λ ) - \zeta(z; Λ ) 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

The Weierstrass p-function is related to the zeta function by

\operatorname{\wp}{(z; Λ )} = - \operatorname{\zeta'}{(z; Λ )}, forany z \in \Complex

The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.

Degenerate case

Consider the situation where one period is real, which we can scale to be

\omega _{1=2\pi} and the other is taken to the limit of \omega _{2 →} i inf ty so that the functions are only singly-periodic. The corresponding invariants are \{g _{2,g} _{3\}=\left\{\tfrac{1}{12},\tfrac{1}{216}\right\}} of discriminant \Delta =0 . Then we have η _{1=\tfrac{\pi}{12}} and thus from the above infinite product definition the following equality:\operatorname{\sigma}{(z; Λ )} =2 e ^{z ^{2/24}} \sin{\left(\tfrac{z}{ 2 }\right)}

A generalization for other sine-like functions on other doubly-periodic lattices is

f(z) = (\pi)/(\omega _{1}) e ^{- ( 4η _{1 /\omega} ^{2} ^{1 )z}} \operatorname{\sigma}{( 2 z; Λ )}