Author Topic: 2.6 #14  (Read 303 times)

Aaron

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2.6 #14
« on: December 05, 2018, 12:18:26 AM »
Can someone show me the details? I am not sure my process

yunhao guan

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Re: 2.6 #14
« Reply #1 on: December 05, 2018, 10:46:47 AM »
Let f(z) = $\frac{z}{z^2+2z+5}\ , $then $z^2+2z+5=0$. We solve the equation, and we get $z= -1 \pm 2i$, only$z= -1+ 2i$ is the upper region.
Therefore, $Res(f, -1+2i)$ = $\dfrac{\sqrt{-1+2i}}{(-1+2i)-(-1-2i)} = \frac{\sqrt{-1+2i}}{4i}\
$ We need to compute $\sqrt{-1+2i}=a+ib$. We square both sides, and we get that $a = \sqrt{\frac{\sqrt{5}-1}{2}} =b^ {-1}$
Thus $I = Re(2\pi\ i$$\frac{a+ib}{4i}) = \frac{\pi\ a}{2}=\frac{\pi}{2}\sqrt{\frac{\sqrt{5}-1}{2}} $
Hope it helps.

Victor Ivrii

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Re: 2.6 #14
« Reply #2 on: December 05, 2018, 02:08:41 PM »
Since $x^2+2x+5$ is not even function one needs to use a "keyhole" contour