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Please help me in finding the image of the strip $\lbrace (x,y) \in\mathbb{C} : mx-\pi<y<mx+\pi,-\infty<x<\infty\rbrace $ under the mapping $w=e^z $ ?

Where $m$ is any real number.

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What's $m$ supposed to be? –  J. M. Aug 26 '12 at 15:17
    
$m$ is any real number, edited –  pritam Aug 26 '12 at 15:23
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1 Answer 1

up vote 1 down vote accepted

$${\mathbb C}\setminus \{0\}$$ as for any complex $(a,b)\neq0$, you can find $\rho>0$ and $\theta\in[-\pi,\pi]$ such that $(a,b)=\rho e^{i\theta}=e(\ln(\rho)+i\theta)$.

So take $x=\ln(\rho)$ and $y$ can be found in $[mx-\pi,mx+\pi]$ such that $y-\theta=2k\pi$ ($k\in\mathbb Z$)

Hence, $(x,y)$ is in your strip and $e^{(x+iy)}=(a,b)$

EDIT : except that your strip doesn't contain $y=mx\pm\pi$. So you will not have all points $\rho.e^{i.(m\rho+\pi)}$ that is some kind of spiral :

$${\mathbb C}\setminus \{\rho.e^{i(m\rho+\pi)}\;|\;\rho\ge0\}$$

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