Sums, Products, and Quotients of Analytic Functions

Theorem

Let $f, g: \C \to \C$ be analytic functions in some open set $U$.

Then $f+g$ and $f g$ are analytic in $U$ as well, and $f/g$ is analytic in $U - \left\{{z : g(z)=0 }\right\}$.

Proof

Since both $f, g$ are analytic, we can construct a complex derivative for $f+g$ as $f' + g'$.

Similarly, the product rule allows a construction of a derivative for $fg$ as $f'g + fg'$.

The product rule allows us to construct a derivative for $f/g$ as $\dfrac{f'g - fg'}{g^2}$, although neither the quotient nor its derivative is defined anywhere $g=0$.

$\blacksquare$