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DeadOriginal
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Homework Statement
Determine ##f(S)## where ##f(z)=e^{\frac{1}{z}}## and ##S=\{z:0<|z|<r\}##.
*Edit: The function f is defined as ##f:\mathbb{C}\rightarrow\mathbb{C}##.
The Attempt at a Solution
I am a little confused as to what this problem is asking me to do. What I did was:
Let ##z=|z|e^{i\theta}##. Then
$$
\begin{align*}
f(z)=e^{\frac{1}{z}}=e^{\frac{1}{|z|e^{i\theta}}}=e^{\frac{\cos\theta-i\sin\theta}{|z|}}
=e^{\frac{\cos\theta}{|z|}}e^{-\frac{i\sin\theta}{|z|}}
=e^{\frac{\cos\theta}{|z|}}\left(\cos(\frac{\sin\theta}{|z|})-i\sin(\frac{\sin\theta}{|z|})\right).
\end{align*}
$$
Can I then say that ##f(S)=e^{\frac{\cos\theta}{|z|}}\left(\cos(\frac{\sin\theta}{|z|})-i\sin(\frac{\sin\theta}{|z|})\right)## for ##S=\{z:0<|z|<r\}## or is there something more to this problem that I am not seeing?
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