Inverse application of the generalized Littlewood theorem concerning integrals of the logarithm of analytic functions: an easy method to establish equalities between different analytic functions

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this Note, we discuss what, in a sense, are inverse applications of this theorem. We first prove an easy Lemma that if two meromorphic on the whole complex plane functions $f(z)$ and $g(z)$ have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large $|z|$, then for some integer $n$, $d^n \ln(f(z)/dz^n = d^n\ln(g(z)/dz^n$. The use of this Lemma enables easy proofs of many identities between elliptic functions and their transformation rules. In particular, we show how for any complex number $a$, $\wp(z)-a$, where $\wp(z)$ is Weierstrass $\wp$-function, can be presented as a product and ratio of three elliptic $θ_1$-functions of certain arguments. We also establish n-tuple rules for elliptic theta-functions and /rho_z(z) functions.