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The derivative of the zeta function is ℘ (), where ℘ is the Weierstrass elliptic function. The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory. Weierstrass eta function
Legendre's relation stated using elliptic functions is = where ω 1 and ω 2 are the periods of the Weierstrass elliptic function, and η 1 and η 2 are the quasiperiods of the Weierstrass zeta function.
Other functions called zeta functions, but not analogous to the Riemann zeta function. Jacobi zeta function; Weierstrass zeta function; Topics related to zeta functions. Artin conjecture; Birch and Swinnerton-Dyer conjecture; Riemann hypothesis and the generalized Riemann hypothesis. Selberg class S; Explicit formulae for L-functions; Trace formula
Moreover, the fact that the set of non-differentiability points for a monotone function is measure-zero implies that the rapid oscillations of Weierstrass' function are necessary to ensure that it is nowhere-differentiable. The Weierstrass function was one of the first fractals studied, although this term was not used until much later. The ...
The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
The uppercase zeta is not used, because it is normally identical to Latin Z. The lower case letter can be used to represent: The Riemann zeta function in mathematics [3] The Hurwitz Zeta Function in mathematics [4] The Weierstrass zeta-function [5] The damping ratio of an oscillating system in engineering and physics [6]
The Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues.
For example, the Weierstrass zeta function associated to a lattice ... since is the zero set of the Weierstrass sigma function (). Mixed Hodge theory for smooth ...