Let $K$ be the field of fractions of $\mathbb{C}[[z]]\otimes_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple criterion which will conclude that $f(zw)$ does not belong to $K?$ I suspect that $f(zw)\in K$ if and only if $f(t)$ is a rational function, i.e. belongs to $\mathbb{C}(t).$ I am especially interested in proving that $e^{zw}\notin K.$