Question: Let the function be holomorphic in the open set . Prove that the function is holomorphic in the set
Side note: the first bar is over the z and the second bar is over the image of f(z)
My idea for the proof:
I want to show that there are first partial derivatives in G* s.t. the partial derivatives are continuous and satisfy the Cauchy Reimann equations
I differentiate this function on G* w.r.t. to x and y to derive the first partial derivatives.
(how i do this is f(x+h,y)-f(x,y) over h and f(x,y+h) -f(x,y) over ih)
However when I do this I get df/dx = du/dx + -i dv/dx
and df/dy = -dv/dy - i du/dy
which implies the function is constant.
Can someone explain to me what I am doing wrong or not seeing?
I think Opalg's solution is correct but requires an advanced theorem; most early in the study of complex analysis have a weaker theorem to work with (easier to prove): if satisfies the CRE's at and (derivatives in the x and y direction) are continuous at then is differentiable at . So you probably have to throw in that and are continuous as well.
Alternatively, you can directly apply the definition of the derivative. So for your problem, take any . Then exists. So, exists (the last step is simply the conjugate of a complex number). Therefore,