Let $U \subset \mathbb{C}$ be open, and let $f: U \to \mathbb{C}$ and $g: U \to \mathbb{C}$ be analytic functions. Define the pointwise sum and pointwise product maps $f+g: U \to \mathbb{C}$ and $fg: U \to \mathbb{C}$ by $(f+g)(z)=f(z)+g(z)$ and $(fg)(z)=f(z)g(z)$ for every $z \in U$. Then $f+g$ and $fg$ are analytic on $U$.
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Moreover, let $V \subset \mathbb{C}$ be open, let $g(U) \subset V$, and let $h: V \to \mathbb{C}$ be analytic. Define the composition map $h \circ g: U \to \mathbb{C}$ by $(h \circ g)(z)=h(g(z))$ for every $z \in U$. Then $h \circ g$ is analytic on $U$.