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Integrate [sin(x) - x]/x^3 from 0 to infinity

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Evaluate the integral from zero to infinity of [sin(x) - x]/x^3 dx using both complex analysis and Ramanujan's master theorem

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To integrate [sin(x) - x]/x^3 from 0 to infinity we can consider the integral from minus infinity to infinity as this will yield twice the desired integral due to the integrand being even. Then we can write this integral as the principal part obtained by removing the contribution to the integral from minus to plus epsilon and taking the limit of epsilon to zero. The integral before taking this limit will be denoted as I(epsilon). We can then evaluate I(epsilon) as follows.

We consider integrating the function [exp(i z) - i z]/z^3 along the contour that starts at -R on the real axis and then moves to to -epsilon, then takes a clockwise half circle with center the origin and radius epsilon it moves to epsilon, from there it moves to R and then with a half ...

Solution Summary

A detailed explanation is given on how to apply complex analysis to solve this problem, also we explain in detail on how Ramanujan's master theorem can be applied here.