Suppose that
is analytic within a region
. If we consider a closed contour
in
and a point
enclosed within
then the Laurent expansion of
about the point
is
given by
where for
Thus for
and for
The Residue Theorem follows from the fact that the coefficient of the
term is
Proof of Laurent's theorem
We consider two nested contours
and
and points
contained in the annular region, and the point
contained within the inner contour. By Cauchy's theorem and the Cauchy Goursat theorem
Integral over 
We begin the proof by rewriting the integrand in the
integral by adding and subtracting
in the denominator,
The factor in square brackets
may be expanded into a geometrical series provided that
Writing up to the
-th term, remainder, we have
Thus, we may substutute this expansion into the original integral over
where
We may estimate the remainder
We note that
which yields the estimate
.
Here the constant factor
follows from the Maximum Modulus Theorem.
Integral over 
We begin the proof by rewriting the integrand in the
integral by adding and subtracting
in the denominator,
The factor in square brackets
may be expanded into a geometrical series provided that
Writing up to the
-th term, remainder, we have
Thus, we may substitute this expansion into the original integral over
where
We may estimate the remainder
We note that
which yields the estimate
.
Here the constant factor
follows from the Maximum Modulus Theorem.
Combining the
and
results
Because the respective remainders for the series representations vanish, we may combine these two results to yield
Finally, the by Cauchy's theorem, the integrals over the contour
and
are equivalent to the integral over any closed contour
which lies in
proving the Laurent's theorem.
It must be mentioned that, like the Taylor's expansion, the Laurent expansion of a function is unique where the function is analytic.