Understanding Dirichlet Integral Proof

[Amok]

I was trying to understand the proof given in this wiki page:

http://en.wikipedia.org/wiki/Dirichlet_integral

But I'm sure their proof is correct because I don't know if the step where he says the sine is the imaginary part of the the exponential and then expands it to the whole of the integral is correct.

$$\int_0^{\infty} e^{-\alpha \omega} Im( e^{i \beta \omega}) = Im( \int_0^{\infty} e^{-\alpha \omega} e^{i \beta \omega})$$

I doesn't make much sense to me. That whole part of the demonstration is very unclear to me.

Anyone know this?

 

[losiu99]

$$e^{-\alpha\omega}$$ is real there, and since $$c \Im \left{a+bi\right}=\Im\left{ac+cbi\right}=cb$$, the equality $$e^{-\alpha\omega}\sin{\beta\omega}=\Im\left{e^{-\alpha\omega}e^{i\beta\omega}\right}$$ is valid. If you are still uncertain of this, the integral can be solved by using twice integration by parts.

 

[Amok]

$$e^{-\alpha\omega}$$ is real there, and since $$c \Im \left{a+bi\right}=\Im\left{ac+cbi\right}=cb$$, the equality $$e^{-\alpha\omega}\sin{\beta\omega}=\Im\left{e^{-\alpha\omega}e^{i\beta\omega}\right}$$ is valid. If you are still uncertain of this, the integral can be solved by using twice integration by parts.

So alpha and omega have to be real for this to work.

 

[Gib Z]

Yes. Also, don't forget your differentials.

 

[Amok]

Yeah, well they do not state that alpha is real in that page. Thanks guys.