Complex Analysis 2.1.2 - 1 If g(w) and f(z) are analytic functions, show that g(f(z)) is also analytic.

Question

Complex Analysis 2.1.2 - 1

If $g(w)$ and $f(z)$ are analytic functions, show that $g(f(z))$ is also analytic.

Answer 1

HINT

One of the definitions of $f$ being analytic is that is satisfies the Cauchy-Riemann equations. Assuming both $g,f$ analytic, can you plug $g(f(z))$ into the Cauchy-Riemann equations and see what happens?

Answer 2

One way is to use the Cauchy Riemann equations. Another way : $\lim\limits_{z \to z_{0}} \frac{g(f(z))-g(f(z_{0}))}{z-z_0}=\lim\limits_{z \to z_{0}} \frac{g(f(z))-g(f(0))}{f(z)-f(0)} \frac{f(z)-f(z_0)}{z-z_0}$. It is not hard to see that this limit is $g'(f(z_0))f'(z_0)$.