Question
Asked Dec 13, 2019
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Use complex contour integration to evaluate the following integrals:

Use an appropriate branch of log z so you can use the following contour. 

dx
x2 +4
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dx x2 +4

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12
‚R
Ye,R = lR+ V%R+ C; + CR
‚R
CR
C:
-R
R
-E
‚R
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12 ‚R Ye,R = lR+ V%R+ C; + CR ‚R CR C: -R R -E ‚R

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Expert Answer

Step 1

Given integral and contour are,

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dx x + 4 Ye,R = 1R+R+C; + Cr CR Ce R -R c.R

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Step 2

To evaluate the given integration, the complex-values function is,

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f(z) =: z' +4 where singularities are + 2i, –2i Then complex integral becomes, -dz z' +4

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Step 3

It can be solved by using residue theorem or Cauchy integral formula. Here z​1⁄2 = e​1⁄2Log z,

then z​1⁄2 has a branch cut. The logarithm branch cut is defined as the negative real axis. So, define it to be the positive real axis. Hence, use keyhole contour, which consists of a small circle about the origin of radius ε say, extending to a line seg...

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