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The Laurent series is a power series representation of a complex function f: ℂ → ℂ. The series is given by

f(z)=\sum_{n=-\infty}^\infty a_n(z-c)^n

with the coefficients an and c given by Cauchy's integral formula:

a_n=\frac{1}{2\pi i} \oint_\gamma \frac{f(z)\,\mathrm{d}z}{(z-c)^{n+1}}

The line integral defining the coefficients must lie on in the region of convergence, which will be between two circles centred at the point c.

Note that unlike the Taylor series, the Laurent series includes terms of negative degree; the sum of such terms is known as the principle part of the Laurent expansion. If the principle part is zero, only the terms of positive degree (known as the analytic part) remain; in this case, the Laurent series is the same as the Taylor series.

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