Dedekind eta function [SEP] named after
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.
Definition.
For any complex number formula_1 with formula_2, let formula_3, then the eta function is defined by,
The notation formula_5 is now standard in number theory, though many older books use "q" for the nome formula_6
Kronecker limit formula
In mathematics, the classical Kronecker limit formula describes the constant term at "s" = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker.
First Kronecker limit formula.
The (first) Kronecker limit formula states that
where
- "E"(τ,"s") is the real analytic Eisenstein series, given by
for Re("s")