This question was part of an assignment and its solution was given but I am having trouble understanding a conclusion.
So, I am asking for help here.
Question is: Let f: $\mathbb{C} \to \mathbb{C}$ be a holomorphic function and let u be it's real part and v be it's imaginary part.Then , for x,y $\in \mathbb{C}$ , $|f'(x+iy)|^2$ is equal to?
So, a line in solution is $|f'(z)|^2 = {u_x}^2+ {v_x}^2$ . why is it true ?
why it shouldn't be $|f'(z)|^2 = {u_x}^2 +{u_y}^2 +{v_x}^2 +{v_y}^2$ ? as in this equation I differentiated u wrt to y and v wrt to x as u and v are both functions of x and y?
Can you please give reason for that
$f'(z)=\lim_{h \to 0} \frac {f(z+h)-f(z)} h$. If you take the limit through real values if $h$ you get $f(z)=u_x+iv_x$ so $|f'(z)|^{2}=u_x^{2}+v_x^{2}$.