Weierstrass functions

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. 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 Weierstrass sigma function associated to a two-dimensional lattice Λ ⊂ C {displaystyle Lambda subset mathbb {C} } is defined to be the product where Λ ∗ {displaystyle Lambda ^{*}} denotes Λ − { 0 } {displaystyle Lambda -{0}} .See also fundamental pair of periods. The Weierstrass zeta function is defined by the sum The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as: 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. The Weierstrass eta function is defined to be