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Homework Statement
Q. Use residues and the contour shown (where R > 1) to establish the integration formula
[tex] \int^{\infty}_{0} \frac{dx}{x^3+1} = \frac{2\pi}{3 \sqrt{3}} [/tex]
The given contour is a segment of an arc which goes from R (on the x-axis) to Rexp(i*2*pi/3)
Homework Equations
The Attempt at a Solution
For the function [itex] f(x) = \frac{dx}{x^3+1} [/tex]
I have worked out that there is only only one pole inside the contour (R >1) which is [tex] x=e^{\frac{i\pi}{3}} [/tex]
So the residue would be
2 * pi * i * Res_([tex] x=e^{\frac{i\pi}{3}} [/tex]) f(x)
But I'm not sure how to do that residue as I've never calculated one with an e in it before... any suggestions? :) Thanks
Homework Statement
p.s sorry I couldn't get the LaTex to show up, someone told me how to do it, but that is obviously not it...
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