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- #1

\[

\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)

\]

when the answer is

\[

-\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 \((*)\)?