Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$.
Basically, they show that the homomorphism given by $g\circ f=[n]$ is well-defined, and they show that $f\circ g=[n]$ as well.
However, they say nothing about the analycity of $g$.
Given an surjective analytic functions $f: X\to Y,h: X\to X$ and an arbitrary function $g. Y\to X$ such that $g\circ f=h$, it is clear that $g$ is continuous, but is it obvious that $g$ is analytic as well ?
In the case of isogenies, the isogeny $f$ is given by an invertible linear transformation $L:V_X\to V_Y$, where $V_X$ is the tangent space of $V_X$ at 0 (and similarly with $Y$). Therefore $g$ is induced by $nL^{-1}$, and so is also analytic.