Editing Laurent series (section)

== Uniqueness ==
Suppose a function <math>f(z)</math> holomorphic on the annulus  <math>r<|z-c|<R</math> has two Laurent series:
<math display="block">f(z) = \sum_{n=-\infty}^{\infty} a_{n} (z-c)^n = \sum_{n=-\infty}^{\infty} b_{n} (z-c)^n.</math>

Multiply both sides by <math>(z-c)^{-k-1}</math>, where k is an arbitrary integer, and integrate on a path γ inside the annulus,
<math display="block">\oint_{\gamma}\,\sum_{n=-\infty}^{\infty} a_{n} (z-c)^{n-k-1}\,dz = \oint_{\gamma}\,\sum_{n=-\infty}^{\infty} b_{n} (z-c)^{n-k-1}\,dz.</math>

The series converges uniformly on <math>r+\varepsilon \leq |z-c| \leq R-\varepsilon</math>, where ''ε'' is a positive number small enough for ''γ'' to be contained in the constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity
<math display="block">\oint_{\gamma}\,(z-c)^{n-k-1}\,dz = 2\pi i\delta_{nk}</math>
into the summation yields
<math display="block">a_k = b_k.</math>

Hence the Laurent series is unique.