Linear differential relations satisfied by Wirtinger integrals

Abstract

We will derive linear differential relations satisfied by Wirtinger integrals by exploiting classical formulas of Jacobi’s theta functions. Although Wirtinger integrals are related to Gauss' hypergeometric functions, we will do that without referring to Gauss' hypergeometric differential equation. We believe that our method to derive them will be applicable to another definite integral defined on the torus which has as integrand a power product of not necessarily four theta functions, and which is not a lift of any definite integral defined on the complex projective line.