Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly.
Let $g(f(z))=s(u(x,y), v(x,y))\,+\,i t(u(x,y), v(x,y))$.
Then we have $\left\{ \begin{array} . \frac{\partial s}{\partial x}=\frac{\partial s}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial s}{\partial v}\frac{\partial v}{\partial x}\\ \frac{\partial t}{\partial y}=\frac{\partial t}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial t}{\partial v}\frac{\partial v}{\partial y}\end{array}\right.$.
If we can show that $\frac{\partial s}{\partial x}=\frac{\partial t}{\partial y}$ and $\frac{\partial s}{\partial y}=-\frac{\partial t}{\partial x}$, we can conclude that the composite $g(f(z))$ is holomorphic. Thus I tried to compute $\frac{\partial s}{\partial x}-\frac{\partial t}{\partial y}$, but I found it difficult to yield the desired result $0$ using Cauchy-Riemann equations of $f$ and $g$. Can I get some advices or hints to proceed more? Thank you in advance.
Best Answer
Since $f$ and $g$ are holomorphic, you know that $\partial s/\partial u=\partial t/\partial v$ and $\partial u/\partial x=\partial v/\partial y$. Therefore the first term in the expression of $\partial s/\partial x$ equals the second one in the expression for $\partial t/\partial y$. Using the other CR equations for $f$ and $g$ you can show that the other two terms are equal as well (the minus signs will cancel)