Suppose that $ f(z) $ is analytic within a region $ {\mathcal {R}} $. If we consider a closed contour $ C $ in $ {\mathcal {R}} $ and a point $ a $ enclosed within $ C $ then the Laurent expansion of $ f(z) $ about the point $ a $ is
given by
$ f(z)=\sum _{n=-\infty }^{\infty }c_{n}(z-a)^{n} $
where for $ n=0,\pm 1,\pm 2,\pm 3,... $
$ c_{n}={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-a)^{n+1}}}\;dw. $
Thus for $ n=0,1,2,... $
$ c_{n}={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-a)^{n+1}}}\;dw $
and for $ n=1,2,3,... $
$ c_{-n}={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-a)^{-n+1}}}\;dw. $
The Residue Theorem follows from the fact that the coefficient of the $ n=-1 $ term is
$ c_{-1}={\frac {1}{2\pi i}}\oint _{C}f(w)\;dw. $
Proof of Laurent's theorem
We consider two nested contours $ C_{1} $ and $ C_{2} $ and points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z
contained in the annular region, and the point
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z = a
contained within the inner contour. By Cauchy's theorem and the Cauchy Goursat theorem
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) = \frac{1}{2 \pi i } \oint_{C_2} \frac{f(w)}{w - z } \; dw - \frac{1}{2 \pi i } \oint_{C_1} \frac{f(w)}{w - z } \; dw.
Integral over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
We begin the proof by rewriting the integrand in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
integral by adding and subtracting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a
in the denominator,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{f(w)}{w - z } = \frac{f(w)}{(w -a -(z -a )) } = \frac{f(w)}{(w -a) } \left[ \frac{1}{1 - \left(\frac{z -a}{w -a } \right)}\right].
The factor in square brackets $ \left[\right] $ may be expanded into a geometrical series provided that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left| \frac{z -a}{w -a } \right| < 1.
Writing up to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N
-th term, remainder, we have
$ {\frac {f(w)}{w-z}}=\sum _{n=0}^{N-1}{\frac {f(w)(z-a)^{n}}{(w-a)^{n+1}}}+{\frac {f(w)}{(w-z)}}\left({\frac {z-a}{w-a}}\right)^{N} $
Thus, we may substutute this expansion into the original integral over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2 \pi i } \oint_{C_2} \frac{f(w)}{w - z } \; dw = \sum_{n=0}^{N-1} \left[\frac{1}{2 \pi i } \oint_{C_2} \frac{f(w )}{(w -a)^{n+1} } \; dw \right] \; (z - a)^{n} + R_N^{+}
where
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_N^{+} = \frac{1}{2 \pi i } \int_{C_2} \frac{ f(w) }{ ( w - z )} \left( \frac{ z - a }{ w - a} \right)^{N} \; dw .
We may estimate the remainder Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_n
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left| R_N^{+} \right| = \left| \frac{1}{2 \pi i } \oint_{C_2} \frac{ f(w) }{ ( w - z )} \left( \frac{ z - a }{ w - a} \right)^{N} \; dw \right| \le \frac{1}{2 \pi} \oint_{C_2} \left| \frac{ f(w) }{ ( w - z )} \left( \frac{ z - a }{ w - a} \right)^{N} \right| \; dw
We note that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): | w - z | = | w - a - (z - a) | \ge | w - a | - | z - a |
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |z - a | < | w - a |
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma = \frac{z -a }{ w -a } < 1
which yields the estimate
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |R_n^{+}| < \frac{1}{2 \pi} M (2 \pi | w - a| ) \frac{\gamma^N}{ | w - a | - |z - a|} =\frac{M\gamma^N |w-a|}{|w-a| - |z -a|} \rightarrow 0 \qquad \mbox{as} \qquad N \rightarrow \infty
.
Here the constant factor $ M $ follows from the Maximum Modulus Theorem.
Integral over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1
We begin the proof by rewriting the integrand in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
integral by adding and subtracting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a
in the denominator,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{f(w)}{z - w } = \frac{f(w)}{(z -a -(w -a )) } = \frac{f(w)}{(w -a) } \left[ \frac{1}{1 - \left(\frac{w -a}{z -a } \right)}\right].
The factor in square brackets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \right]
may be expanded into a geometrical series provided that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left| \frac{w -a}{z -a } \right| < 1.
Writing up to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N
-th term, remainder, we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\frac{f(w)}{w - z } = \sum_{n=0}^{N-1} \frac{f(w) (z -a)^{-n} }{(w -a)^{-n+1}} + \frac{ f(w) }{ ( z - w )} \left( \frac{ w - a }{ z - a} \right)^{N}
Thus, we may substitute this expansion into the original integral over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): - \frac{1}{2 \pi i } \oint_{C_1} \frac{f(w)}{w - z } \; dw = \sum_{n=1}^{N-1} \left[\frac{1}{2 \pi i } \int_{C_1} \frac{f(w )}{(w -a)^{-n+1} } \; dw \right] \; (z - a)^{-n} + R_N^{-}
where
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_N^{-} = \frac{1}{2 \pi i } \int_{C_1} \frac{ f(w) }{ ( w - z )} \left( \frac{ w- a }{ z - a} \right)^{N} \; dw .
We may estimate the remainder Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_n
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left| R_N^{-} \right| = \left| \frac{1}{2 \pi i } \oint_{C_1} \frac{ f(w) }{ ( w - z )} \left( \frac{ w - a }{ z- a} \right)^{N} \; dw \right| \le \frac{1}{2 \pi} \oint_{C_1} \left| \frac{ f(w) }{ ( z - w)} \left( \frac{ w - a }{ z - a} \right)^{N} \right| \; dw
We note that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): | z - w | = | z - a - (w - a) | \ge | z - a | - | w - a |
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |w - a | < | z - a |
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma = \frac{w -a }{ z -a } < 1
which yields the estimate
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |R_n^{-}| < \frac{1}{2 \pi} M (2 \pi | w - a| ) \frac{\gamma^N}{ | z - a | - |w - a|} =\frac{M\gamma^N |w-a|}{|z-a| - |w -a|} \rightarrow 0 \qquad \mbox{as} \qquad N \rightarrow \infty
.
Here the constant factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M
follows from the Maximum Modulus Theorem.
Combining the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_1
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
results
Because the respective remainders for the series representations vanish, we may combine these two results to yield
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) = \frac{1}{2 \pi i } \left\{ \sum_{n=0}^{\infty} \left[ \oint_{C_2} \frac{f(w)}{(w - a)^{n+1} } \; dw \right] (z - a)^{n} + \sum_{n=1}^{\infty} \left[ \oint_{C_1} \frac{f(w)}{(w - a)^{-n+1} } \; dw \right] (z - a)^{-n} \right\}.
Finally, the by Cauchy's theorem, the integrals over the contour $ C_{1} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C_2
are equivalent to the integral over any closed contour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C
which lies in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal R}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) = \frac{1}{2 \pi i } \left\{ \sum_{n=0}^{\infty} \left[ \oint_{C} \frac{f(w)}{(w - a)^{n+1} } \; dw \right] (z - a)^{n} + \sum_{n=1}^{\infty} \left[ \oint_{C} \frac{f(w)}{(w - a)^{-n+1} }\; dw \right] (z - a)^{-n} \right\} = \sum_{n=-\infty}^{\infty} \left[ \frac{1}{2 \pi i } \oint_{C} \frac{f(w)}{(w - a)^{n+1} } \; dw \right] (z - a)^{n}
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.