Example 5.4.3
We wish to compute the integral
.
The denominator vanishes at two points,
and
, both inside the contour. We will decompose this contour into two Jordan
curves by the following way: draw the diameter of the circle which coincides
with the x-axis. Denote:
-
= the upper half circle together with the diameter, oriented positively.
-
= the lower half circle together with the diameter, oriented positively.
Then
(the variable
``travels'' twice on the diameter, but in opposite directions).
Figure 5:
 |
Therefore we have:
![$\displaystyle = 2 \pi i \cdot \left[ \frac {1}{z+i} \right]_{z=i} + 2 \pi i \cdot \left[ \frac {1}{z-i} \right]_{z=-i}$](/images/maths/complex-analysis/cauchy-tf/img904.png)
