(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate [tex]I=P\int^{\infty}_{- \infty} \frac{e^{ikx}}{x} dx[/tex]

2. Relevant equations

[tex]I=P\int^{\infty}_{- \infty} \frac{f(x)}{x-x_0} dx = i \pi f(x_0) + 2 \pi i \sum a_{-1}(z_+)[/tex]

3. The attempt at a solution

According to Maple the solution is [tex]2i\pi[/tex]. Now if I try to calculate it using the above formula, I find [tex]f(x_0)=e^0=1[/tex]

and since f(x)/x doesn't have any poles in the upper halfplane the sum of the residues is zero. This leads to

[tex]I=P\int^{\infty}_{- \infty} \frac{f(x)}{x-x_0} dx = i \pi[/tex]

Where did I go wrong?

Also, to use this formula for the integral the line integral over the upper halfplane must be zero. So to prove this you have to calculate: [tex]\lim_{R\rightarrow \infty} R f(R)[/tex] This is not zero. I'm confused now...

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# Homework Help: Cauchy Principal Value integral

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