Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.