2019/12/31 by Andrew Bakan, Håakan Hedenmalm, Håkan Hedenmalm · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic and geometric function theory #Combinatorics #Complex plane #Holomorphic function #Hyperbolic function #Logarithm #Mathematical analysis #Mathematics #Physics #Theta function #math.CA #msc:33C05 #msc:33E05
paper · pdf · doi:10.1007/s40315-020-00332-x
published in Computational Methods and Function Theory 20(3-4), 591-621 (Springer Science+Business Media) · 74 pages
openalex publication_date 2020/07/29 · arxiv created 2021/08/24 · arxiv updated 2021/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let Θ3 (z):= ∑n∈ℤ exp (i πn2 z) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane ℍ, and takes positive values along the positive imaginary axis. We define its logarithm logΘ3(z) which is uniquely determined by the requirements that it should be holomorphic in ℍ and real-valued on the positive imaginary axis. We derive an integral representation of logΘ3 (z) when z belongs to the hyperbolic quadrilateral F||\square, consisted of all those z ∈ ℍ which satisfy -1 ≤ Re z ≤ 1, |2 z - 1| > 1 and |2 z + 1| > 1. Since every point of ℍ is equivalent to at least one point in F||\square under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of ℍ via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions Θ4 and Θ2 may be conveniently expressed in terms of logΘ3 via functional equations, and hence get controlled as well. Our approach is based on a study the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This connects with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.