If a function fz has an isolated singular point at z0 then it can be

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If a function f(z) has an isolated singular point
at z0 then it can be represented by a Laurent
series, and the coefficient b1 of the 1/(z- z0)
term is called the residue of f(z) at z0.
As shown in the section on the Laurent series,
contour around z0 and lying in the region of
convergence for the Laurent series. Sometimes
the residue is not easy to calculate ( see
example 4 on pages 181 182 ). But there are
cases called poles where we can do it easily.
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If bn 0 for n gt m we say f(z) has a mth order
pole. Consider for example
In general z0 is a mth order pole of f(z) if
only if f(z) ?(z)/(z?z0)m and the residue is
?(z0) if m 1 and it is ?m-1(z0) / (m-1)! if m gt
2.
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If all we needed were residues, we are finished.
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A good problem to test your understanding of the
above material is to show that if you just want
the values of the 2 residues of H(s) shown
below, you do not have to find a, b, c.