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## Homework Statement

The following integral is given:

[tex]\int_{0}^{\infty} e^{ikx} dx[/tex]

k & x are real.

## Homework Equations

We know that:

[tex]\int_{-\infty}^{\infty} e^{ikx} dx=2\pi \delta(k)[/tex]

## The Attempt at a Solution

The answer is:

[tex]\mathcal{I}=\pi \delta(k) + i \frac{1}{k}[/tex]

I could easily prove the real part with the formula in

**2**, but couldn't be persuaded why the imaginary part is so (it barely makes sense). I tried:

[tex]\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)[/tex]

But [itex]\lim_{M\rightarrow \infty} \cos(kM)[/itex] is quite meaningless and I can't see why the whole expression should be k^{-1}...

I'll appreciate any help,

Thanks!

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