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MOTIVATION

As an example, consider the contour integral
:\oint_C {e^z \over z^5}\,dz
where ''C'' is some Jordan Curve about 0.

Let us evaluate this integral without using standard integral theorems that may be available to us. Now, the Taylor Series for ''e''''z'' is well-known, and we substitute this series into the integrand. The integral then becomes:
:\oint_C {1 \over z^5}\left(1+z+{z^2 \over 2!} + {z^3\over 3!} + {z^4 \over 4!} + {z^5 \over 5!} + {z^6 \over 6!} + \ldots ight)\,dz
Let us bring the 1/''z''5 term into the series, and so, we obtain
:\oint_C {1 \over z^5}+{z \over z^5}+{z^2 \over 2!\;z^5} + {z^3\over 3!\;z^5} + {z^4 \over 4!\;z^5} + {z^5 \over 5!\;z^5} + {z^6 \over 6!\;z^5} + \ldots\,dz
:\oint_C {1 \over\;z^5}+{1 \over\;z^4}+{1 \over 2!\;z^3} + {1\over 3!\;z^2} + {1 \over 4!\;z} + {1\over\;5!} + {z \over 6!} + \ldots\,dz

The integral now collapses to a much simpler form. Recall
:\oint_C {1 \over z^a} \,dz=0,\quad a \in \mathbb{R}, \ a
e 1

So now the integral around ''C'' of every other term not in the form ''cz''−1 becomes zero, and the integral is reduced to
:\oint_C {1 \over 4!\;z} \,dz={1 \over 4!}\oint_C{1 \over z}\,dz={1 \over 4!}(2\pi i) = {\pi i \over 12}

The value 1/4! is known as the ''residue'' of ''e''''z''/''z''5 at ''z''=0, and is notated as
:\mathrm{Res}_0 {e^z \over z^5},\ \mathrm{or}\ \mathrm{Res}_{z=0} {e^z \over z^5},\ \mathrm{or}\ \mathrm{Res}(f,0).


CALCULATING RESIDUES

Suppose a defined (at least) on ''D''. The residue Res(''f'', ''c'') of ''f'' at ''c'' is the coefficient ''a''−1 of (''z'' − ''c'')−1 in the Laurent Series expansion of ''f'' around ''c''. At a Simple Pole , the residue is given by:

:\operatorname{Res}(f,c)=\lim_{z o c}(z-c)f(z).
According to the integral formula given in the Laurent Series article we have:

:\operatorname{Res}(f,c) =
{1 \over 2\pi i} \int_\gamma f(z)\,dz

where γ traces out a circle around ''c'' in a counterclockwise manner. We may choose the path γ to be a circle in radius ε around ''c'' where ε is as small as we desire.

The residue of a function ''f''(''z'')=''g''(''z'')/''h''(''z'') at a Simple Pole ''c'', where ''g'' and ''h'' are holomorphic functions in a neighborhood of ''c'' with ''h(c)'' = 0 and ''g(c)'' ≠ 0 is given by
:\operatorname{Res}(f,c) = rac{g(c)}{h'(c)}.

More generally, the residue of ''f'' around ''z'' = ''c'', a Pole of order ''n'', can be found by the formula:

: \mathrm{Res}(f,c) = rac{1}{(n-1)!} \cdot \lim_{z o c} rac{d^{n-1}}{dz^{n-1}}\left( f(z)\cdot (z-c)^{n} ight).