2
$\begingroup$

I am required to prove the following:

For any real number $k$, prove that the exponential function $e^z$ is a bijection ($z$ is a complex number) from the strip $a < im z \leq k+2pi$ to the complex plane minus the point $0$, $\Bbb C - \{0\}$.

Any hints please? Thanks!

  • 0
    It can't be *any* $k\in \mathbb{R}$, because what if $k=3$ and $im(z)=2$?2012-08-26

2 Answers 2

1

Hint To solve $e^{x+iy}=\omega$, write $\omega$ in trigonometric form and solve.

  • 0
    thank you, but i don't get how is my domain being defined? Also, does this 'absolute value' you are talking about |w| = e^x?2012-08-26
1

A hint: Don't solve equations, but investigate what the exponential function $z=x+iy \ \mapsto\ e^z=e^x\cdot e^{iy}$ does to horizontal lines $g_v:\quad y:= v\ (={\rm const.})\ , \quad -\infty and to vertical lines $h_u:\quad x:= u\ (={\rm const.}), \quad -\infty