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I need to find the cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$ ? I'm think of rewriting in terms of $e^{i\theta}$ and try to rewrite as contour integral?

Need some aid on how to calculate this integral please. Thank you for help

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up vote 2 down vote accepted

All that is needed to translate the integrand to obtain

\begin{equation} \mathrm{PV}\int_{-\infty}^{\infty}\frac{\sin(x+a)}{x}\,dx = \mathrm{PV}\int_{-\infty}^{\infty}\frac{\sin x \cos a + \cos x \sin a }{x}\,dx. \end{equation}

We already know that

\begin{equation} \int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx = \pi \qquad\text{and}\qquad \mathrm{PV}\int_{-\infty}^{\infty}\frac{\cos x}{x}\,dx = 0. \end{equation}

Even if you did not know these results, they are easier to deal with than the original integral. Then the rest follows immediately.

Of course, you can evaluate it manually by contour integration. It will be greatly helpful to recall how we evaluated the

$$ \int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx = \pi. $$

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