# Some complex Fourier-like infinite integral

1. Mar 11, 2007

### Ahmes

1. The problem statement, all variables and given/known data
The following integral is given:
$$\int_{0}^{\infty} e^{ikx} dx$$
k & x are real.

2. Relevant equations
We know that:
$$\int_{-\infty}^{\infty} e^{ikx} dx=2\pi \delta(k)$$

3. The attempt at a solution
$$\mathcal{I}=\pi \delta(k) + i \frac{1}{k}$$
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:
$$\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)$$
But $\lim_{M\rightarrow \infty} \cos(kM)$ is quite meaningless and I can't see why the whole expression should be k^{-1}...

I'll appreciate any help,
Thanks!

Last edited: Mar 11, 2007
2. Mar 11, 2007