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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.

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Step 1

Given integral and contour are,

Step 2

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

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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