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## Homework Statement

Find a Laurent series of ##f(z)=ze^{1/z}## in powers of ##z-1##. Is there an easier way to go about this as this is not a typical expansion I see on textbooks. It seems that my incomplete solution is too complicated. Please help, exam is in two days and I am working on past exams. Worked out solutions are welcome, also. P.S. it may seem like I'm cramming, but it's not under my control -

## Homework Equations

## The Attempt at a Solution

Since ##f(z)## is entire, then it is analytic in any annular domain. So let's pick the domain ##\frac{1}{2} \lt \vert z-1\vert \lt 3##. By Laurent's theorem the coefficients

\begin{align}a_k=\frac{1}{2\pi i}\int_{c}\frac{(1+e^{it})e^{\frac{1}{1+e^{it}}}}{e^{it\cdot k}}\end{align}

where ##C## is the contour ##z(t)=1+e^{it}## and hence, ##dz=ie^{it}dt##. Note that the contour ##C## is within our selected annular domain as it should be according to the theorem.

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