Exercise · ZW-LAURENT-SERIES-005 · ★★☆ Intermediate
Compute \(\oint_{|z|=1}e^{3/z}\,dz\), with the circle traversed once counterclockwise.
Choose one option.
Show answer
- D. \(6\pi i\)
Explanation
The coefficient of \(z^{-1}\) in \(e^{3/z}=\sum 3^k z^{-k}/k!\) is \(3\), and \(\oint z^{n}dz\) is nonzero only for \(n=-1\) (it equals \(2\pi i\)). Result: \(6\pi i\).
Review the concept: Laurent series
The Laurent series of a function \(f\) holomorphic on an annulus \(r<|z-z_0|<R\) is its expansion in integer powers, positive and negative, of \(z-z_0\), with coefficients \(a_n\).
\[f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n\]
More practice problems: laurent series
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