There is a type of problems in my course in Complex analysis that I don't fully understand them.
Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and bijective domain of this function.
I'll show you my solution, and then ask the question.
Since $z\in\mathbb{C}$ we can present it in the form of $z=x+iy$, where $x,y\in\mathbb{R}$. Then it's easy to present $f(z)$ it in the form of $f(x,y)=u(x,y)+iv(x,y)$.
$$f(x,y)=(x+iy)^2=x^2-y^2+2iyx$$ So $u(x,y)=x^2-y^2$ and $v(x,y)=2yx$ Then we can check analyticity of function by substituting these functions into Cauchy–Riemann equations .
To check bijective of function $f(z)$ I used this theorem:
Function is bijective if and only if it is invertible.
So I used Inverse function theorem to find point in neighbourhood of which the function is invertible.
For this problem I got that function has the inverse in neighbourhoods of all points except neighbourhood of point $(0,0)$ (Jacobian equal $0$ in this point)
So my question is
- I found properties of function $f(z)$ in the neighborhood of any point, but (as I think) I do not answer for the question of problem. Because knowing the properties in the neighborhood of any point does not give anything about the bijective domain. (You can see it here in "Example" section.) I would be grateful if you would explain to me what to do in such a situation.
Update
I took into account remarks in comments: I need a maximum domain of bijectivity.