1. A singular point is a point where the function is not regular. It is the point where the denominator is 0 in a rational function.
Basic formula
Residue formula For is the singularity of
with pole of order
,
(a) The points where the denominator is 0 are . Then we find the residue of
. Expand
using partial fraction expansion. Then
(b) The points where the denominator is 0 are
. Then we find the residue of
. Expand
using partial fraction expansion.
(c) The points where the denominator is 0 are
. Then find the residue of
. We can not use the partial fraction expansion on
. So, we use Taylor expansion on
at
. Then divide. Let
. Then
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(d) The points where the denominator is 0 are . Then find the residue of
. Expand
using the partial fraction expansion. Then
2. Residue formula
The function is analytic on and inside the single closed curve
and monovalent, except for the finite number of points
inside it. Then
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3.
(a) The circle contains the singularity
. So find
. Then
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(b) The circle contains the singularity
. Here
is found by A. Thus, we find
.
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Thus by the residue theorem,
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(c) The circle contains all singularity
. Note that
are already found by A. So, we find
.
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Then by the residue theorem,
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