In the process of calculating the integral [itex] \int_0^{2\pi}\frac{\sin{x} \cos{x}}{\sin{x}+\cos{x}}dx [/itex] by contour integration,I got the following:
[itex]
-\frac{1}{2}[ \LARGE{\oint} \large{\frac{z^2}{(1-i)z^2+i+1}}dz-\LARGE{\oint}\large{\frac{z^{-2}}{(1-i)z^2+i+1}}dz] [/itex]
Where the contour of integration for both integrals is the unit circle centered at the origin. The poles are at [itex]z=\pm i \sqrt{i}=\pm \frac{\sqrt{2}}{2}(1-i) [/itex]. As you can see, [itex] |z|=1 [/itex] and so they're on the contour.
My question is,how should I treat such poles?
Should I exclude them and calculate the integrals as [itex] \pi i \sum_i r_i [/itex] or should include them and use [itex] 2 \pi i \sum_i r_i [/itex] ?
How should I decide with what sign each of the residues should appear in the calculation of integrals?
Thanks
[itex]
-\frac{1}{2}[ \LARGE{\oint} \large{\frac{z^2}{(1-i)z^2+i+1}}dz-\LARGE{\oint}\large{\frac{z^{-2}}{(1-i)z^2+i+1}}dz] [/itex]
Where the contour of integration for both integrals is the unit circle centered at the origin. The poles are at [itex]z=\pm i \sqrt{i}=\pm \frac{\sqrt{2}}{2}(1-i) [/itex]. As you can see, [itex] |z|=1 [/itex] and so they're on the contour.
My question is,how should I treat such poles?
Should I exclude them and calculate the integrals as [itex] \pi i \sum_i r_i [/itex] or should include them and use [itex] 2 \pi i \sum_i r_i [/itex] ?
How should I decide with what sign each of the residues should appear in the calculation of integrals?
Thanks
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