- #1

ELESSAR TELKONT

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**[SOLVED] Complex Variables. Problem about complex sine.**

## Homework Statement

Proof that the function

[tex]\begin{displaymath}

\begin{array}{cccc}

f: &A=\left\{z\in\mathbb{C}\mid-\frac{\pi}{2}<\Re z<\frac{\pi}{2}\right\} &\longrightarrow &B=\mathbb{C}-\left\{z\in\mathbb{C}\mid \Im{z}=0,\,\left\vert\Re z\right\vert\geq 1\right\}\\

&z &\mapsto &\sin(z)

\end{array}

\end{displaymath}[/tex]

is a biyection.

## Homework Equations

## The Attempt at a Solution

I have already proven (on the previous exercise of my book) that the complex sine is inyective on a vertical band of the complex plane of width [tex]\pi[/tex]. Then I have only to proof that the function is surjective or that there exists left inverse (more difficult since its multivaluated in general).

My problem is that I have no idea how to proof that since I have saw that the complex sine maps vertical lines in the complex plane to hyperbolas and horizontal ones to ellipses. Then I think it's imposible that the sine could map the set [tex]A[/tex] to the set [tex]B[/tex] one-one because the hyperbolas and the ellipses I have mentioned get out from [tex]B[/tex].