Editing Laurent series (section)

==Definition==
The Laurent series for a complex function <math>f(z)</math> about an arbitrary point <math>c</math> is given by{{sfn|Ablowitz|Fokas|2003|p=128|ps=none}}<ref>{{citation | last=Folland | first=Gerald B. | author-link=Gerald Folland | title=Fourier analysis and its applications | publisher=Wadsworth & Brooks/Cole | publication-place=Pacific Grove, Calif | date=1992 | isbn=978-0-534-17094-3 |page=395}}</ref>
<math display="block">f(z) = \sum_{n=-\infty}^\infty a_n(z-c)^n,</math>
where the coefficients <math>a_n</math> are defined by a [[contour integral]] that generalizes [[Cauchy's integral formula]]:
<math display="block">a_n =\frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{(z-c)^{n+1}} \, dz.</math>

The path of integration <math>\gamma</math> is counterclockwise around a [[Jordan curve]] enclosing <math>c</math> and lying in an [[annulus (mathematics)|annulus]] <math>A</math> in which <math>f(z)</math> is [[holomorphic function|holomorphic]] ([[analytic function|analytic]]). The expansion for <math>f(z)</math> will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled <math>\gamma</math>. When <math>\gamma</math> is defined as the [[Circle#Complex_plane|circle]] <math> |z-c| = \varrho</math>, where <math>r < \varrho < R</math>, this amounts
to computing the complex [[Fourier coefficients]] of the restriction of <math>f</math> to <math>\gamma</math>.{{sfn|Ablowitz|Fokas|2003|pp=196-197|ps=none}} The fact that these integrals are unchanged by a deformation of the contour <math>\gamma</math> is an immediate consequence of [[Green's theorem]]. 

One may also obtain the Laurent series for a complex function <math>f(z)</math> at <math> z = \infty</math>. However, this is the same as when <math> R \rightarrow \infty</math>. 

In practice, the above integral formula may not offer the most practical method for computing the coefficients
<math>a_n</math> for a given function <math>f(z)</math>; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is [[Unique (mathematics)|unique]] whenever
it exists, any expression of this form that equals the given function 
<math>f(z)</math> in some annulus must actually be the Laurent expansion of <math>f(z)</math>.