Math 6420 • PDE

Let \(D_1\) and \(D_2\) be two disjoint domains, whose boundaries share a common curve
\(\Gamma\). Let
\begin{itemize}
\item \(f(z)\) be analytic in \(D_1\) and continuous in \(D_1\cup\Gamma\)
\item \(g(z)\) be analytic in \(D_2\) and continuous in \(D_2\cup\Gamma\)
\item \(f(z)=g(z)\) for \(z\in\Gamma\)
\end{itemize}
Show that the function
\[
  H(z)=
  \begin{cases}
    f(z), & z\in D_1,\\
    f(z)=g(z), & z\in\Gamma,\\
    g(z), & z\in D_2
  \end{cases}
\]
is analytic in \(D=D_1\cup\Gamma\cup D_2\).

[Hint: Use the Morera theorem.]