[Math] Derivative of complex conjugate

Question

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial derivative of the complex conjugate of f with respect to x equal to the complex conjugate of the partial derivative of f with respect to x ? Any help would be greatly appreciated!

Answer 1

Always.

Let $f(x,y)=u(x,y)+iv(x,y)$ with real functions $u$ and $v$. Then $f_x(x,y)=u_x(x,y)+iv_x(x,y)$. For the function $$g(x,y):=\overline{f(x,y)}=u(x,y)-iv(x,y)$$ we obtain $$g_x(x,y)=u_x(x,y)-iv_x(x,y)=\overline{f_x(x,y)}\ .$$