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knowlewj01
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Homework Statement
evaluate the integral:
[itex]I_1 =\int_0^\infty \frac{dx}{x^2 + 1} [/itex]
by integrating around a semicircle in the upper half of the complex plane.
Homework Equations
The Attempt at a Solution
first i exchange the real vaiable x with a complex variable z & factorize the denominator. Also, the contour of integration is a semicircle with radius= infinity
[itex]I_2 = \int_{-\infty}^{\infty}\frac{dz}{(z+i)(z-i)}[/itex]
the contour contains only the pole in the upper half, so from residue theorem we know:
[itex]I_2 = 2\pi i R(i)[/itex]
where R(i) is the residue at the point z=i
R(i) = 1/(i+i) = 1/2i
Hence [itex]I_2 = \pi[/itex]
Now, i know the answer to the original integral is supposed to be pi/2.
Can i say that: because the original limits range from 0 to infinity, and i have integrated twice this amount, my answer should be divided by 2? or is this reasoning flawed?