Using Cauchy Riemann equations

Question

If i had some function $g(f(z)) $ and wanted to use the Cauchy Riemann equations to check if it's entire how would I get $ g(f(z))$ in the form $u(x,y)+iv(x,y)$ if the functions were real we would usually use the chain rule? For example how would I use the Cauchy Riemann equations to check of something like $sin(sin(z))$

Answer 1

Let $h=g(f(z))=h_r+ih_i$ and let $w=f(z)=u(x,y)+iv(x,y)$. Then, we have from the chain rule

$$\frac{h_r}{\partial x}=g_r'u_x-g_i'v_x$$

and

$$\frac{\partial h_i }{\partial y}=g_i'u_y+g_r'v_y $$

If $f$ is analytic, then $u_x=v_y$ and $u_y=-v_x$. Hence, if $g$ is differentiable and $f$ is analytic, then $\frac{\partial h_r}{\partial x}=\frac{\partial h_i}{\partial y}$.

Can you show that the other CRE is satisfied?