If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.

Question

If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.

Let $f(z)=u+iv$, then $f^2=u^2-v^2+2iuv=X+iY$, say

then by CR equation due to analyticitiy, it can be shown that CR equations hold for $f^2$.

But how to show that $f^2(z)$ is analytic in D?

Answer 1

Suppose that $f$ is analytic in $D$. Then for any $z\in D$, \begin{align*} & \mathop {\lim }\limits_{h \to 0} \frac{{f^2 (z + h) - f^2(z)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{(f(z + h) - f(z))(f(z + h) + f(z))}}{h} \\ & = \mathop {\lim }\limits_{h \to 0} \frac{{f(z + h) - f(z)}}{h}\mathop {\lim }\limits_{h \to 0} (f(z + h) + f(z)) = f'(z) \cdot 2f(z), \end{align*} i.e., $f^2$ is also analytic in $D$ and $(f^2(z))'=2f'(z)f(z)$.