31. Use a rectangular contour to evaluate the integrals:

(a) MATH $0<\func{Re}a<b$

The upper side of the rectangle should be at $y=2\pi /b$ (for real $b).$ Then on the upper side:
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Then around the whole rectangle:
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Along the end at $x=R$, with $a=\alpha +i\gamma $
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provided that $\func{Re}a<b.$

Along the end at $x=-R:$
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provided that $\func{Re}a>0.$ Thus we have:
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Ch2pr31a__15.png
Now the integrand has a pole where
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or
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which is inside the contour. The residue there may be found from method 4:
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and so the integral is:
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and thus
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The result is real, as expected.

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