# Complex integration

#### dwsmith

##### Well-known member
Consider
$\int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx$
where $$a,b>0$$. The poles are $$x=\pm b$$ which are on the x axis. Usually, if the poles are on the x axis, I use that the integral is
$2\pi i\sum_{\text{UHP}}\text{Res} + \pi i\sum_{\text{x axis}}\text{Res}\quad (*)$
which works in this problem http://mathhelpboards.com/analysis-50/integral-=-2pi-sum-res-uhp-pi-i-sum-res-real-axis-7576.html
However, if I use this formula on the integral above, I get the answer to be
$-\frac{\pi}{b}\sin(ab)$
$-\frac{2\pi}{b}\sin(ab)$
which would indicate $$2\pi i$$ times the sum of the residual on the x axis. What is going wrong and when can and cannot I use the formula $$(*)$$?

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
$$\displaystyle PV \int^{\infty}_{-\infty} \frac{e^{iaz}}{z^2-b^2}\,dz$$

The function has only poles on the real axis at $$\displaystyle z=\pm b$$

so that becomes

$$\displaystyle PV \int^{\infty}_{-\infty}\frac{e^{iaz}}{z^2-b^2}\,dz= \pi i \lim_{z\to b}\, (z-b)\frac{e^{iaz}}{z^2-b^2}+\pi i \lim_{z\to -b}\, (z+b)\frac{e^{iaz}}{z^2-b^2}=\frac{\pi i e^{iab}}{2b}-\frac{\pi i e^{-iab} }{2b}=\frac{\pi i}{2b }(e^{iab}-e^{-iab})$$

which is equal to $$\displaystyle -\frac{\pi \sin(ab)}{b}$$ . As yours .

Note this the Principle value of the integral >

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#### dwsmith

##### Well-known member
The answer is $$-\frac{2b}{\pi}\sin(ab)$$ which isn't what we both have.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
The answer is $$-\frac{2b}{\pi}\sin(ab)$$ which isn't what we both have.
I cannot see how that would be the correct answer. why so sure ?

#### Random Variable

##### Well-known member
MHB Math Helper

But just so you're aware, that formula is generally only applicable when the the poles on the real axis are simple poles.

But it is also applicable if none of the Laurent expansions about the poles on the real axis have terms of negative even power.

That's why $\displaystyle \text{PV} \int_{-\infty}^{\infty} \frac{1}{x^{3}}\ dx = 0$.

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#### dwsmith

##### Well-known member
@Random my professor says it depends on how we construct the contour around the poles. How true is that? Shouldn't the integral be the same?

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
If we avoid the poles then the integral along any closed smooth path is zero.