Kronecker limit formula
In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series 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 Kronecker limit formula states that
where
for Re > 1, and by analytic continuation for other values of the complex number s.
So the Eisenstein series has a pole at s = 1 of residue π, and the Kronecker limit formula gives the constant term of the Laurent series at this pole.Second Kronecker limit formula
The second Kronecker limit formula states that
where
- u and v are real and not both integers.
- q = e2π i τ and qa = e2π i aτ
- p = e2π i z and pa = e2π i az
for Re > 1, and is defined by analytic continuation for other values of the complex number s.
