Let $D_1$ and $D_2$ be two compact sets in $\Bbb C$, $D_1\cap D_2=\emptyset$, and $ f\colon \Bbb C\setminus(D_1\cup D_2)\to\Bbb C$ be a holomorphic function. Show that there exist two holomorphic functions $ f_1\colon \Bbb C\setminus D_1\to\Bbb C$ and $ f_2\colon \Bbb C\setminus D_2\to\Bbb C$ such that $$f=f_1+f_2$$ for all $z\in C\setminus(D_1\cup D_2)$.
I think about Cauchy integral formula. A similar question : Lemma 1 Math 246A, Notes 4: singularities of holomorphic functions
For $r$ small take some closed loop $C_r$ enclosing $D_1$ at a distance $< r$ and let $$ g(z)=\frac1{2i\pi} \left[\;\,\int\limits_{|z|=1/r} \frac{f(s)}{s-z}ds -\int\limits_{C_r} \frac{f(s)}{s-z}ds\right] $$ $g$ is analytic in the region between $C_r$ and $|z|=1/r$ and it doesn't depend on $r$, thus it is analytic on $\Bbb{C}-D_1$. $$ f(z)=g(z)+(f(z)-g(z)) $$ Do you see why $f(z)-g(z)$ is analytic on $\Bbb{C}-D_2$ ?