Asymptotic Values of Analytic Functions Connected with a Prime End of a Domain

In 1954 M. Heins proved that for any analytic set $A$, containing the infinity, there exists an entire function with asymptotic set $A.$ In the article we prove the following analog of Heins's theorem: for a multi-connected planar domain $D$ with an isolated boundary fragment, an analytic set $A$, $\infty\in A$, and a prime end of $D$ with impression $p$ there exists an analytic in $D$ function $f$ such that $A$ is the set of asymptotic values of $f$ connected with $p$.