Integral of e^iwt: Solving for \omega & t Values

[skateboarding]

For particle location, perturbation theory, etc, I see the following integral.

\LARGE \int_0^t { e^{i\omega t^'}}dt^'

Where \omega is some constant, or frequency. It says in my text that this is equal to 0 if \omega is not close to 0. My logic leads me to think that when \omega is large, the function oscillates many times between 0 and t, so it's integral is 0. However, when I carry out the integral explicitly, it is less clear.

\LARGE {\int_0^t {e^{i\omega t^'}}dt^'} = \frac{e^{i\omega t}}{i\omega} = \cos{\omega t} + i\sin{\omega t} - 1}

My question is, from the expression above, how to I show that this integral is 0 or close to zero? I think it depends on the values chosen for \omega and t. If t is greater than \omega, the integral should give a small number, and if t is close to \omega, the integral should give some non zero value. Any help would be appreciated.

 

[Tangent87]

You made a mistake in evaluating the integral, it should be:

<br /> \LARGE {\int_0^t {e^{i\omega t^&#039;}}dt^&#039;} = \frac{1}{i\omega}\left(e^{i\omega t}-1\right)

Do you see why?

 

[skateboarding]

You made a mistake in evaluating the integral, it should be:

<br /> \LARGE {\int_0^t {e^{i\omega t^&#039;}}dt^&#039;} = \frac{1}{i\omega}\left(e^{i\omega t}-1\right)

Do you see why?

Oops, forgot the denominator. I think that for large t, This integral becomes a delta function. Still working on it. Thanks.

 

[skateboarding]

fixed it

For particle location, perturbation theory, etc, I see the following integral.

\LARGE \int_0^t { e^{i\omega t^&#039;}}dt^&#039;

Where \omega is some constant, or frequency. It says in my text that this is equal to 0 if \omega is not close to 0. My logic leads me to think that when \omega is large, the function oscillates many times between 0 and t, so it's integral is 0. However, when I carry out the integral explicitly, it is less clear.

\LARGE {\int_0^t {e^{i\omega t^&#039;}}dt^&#039;} = e^{i\omega t}}{i\omega} = \frac {\cos{\omega t} + i\sin{\omega t} - 1

My question is, from the expression above, how to I show that this integral is 0 or close to zero? I think it depends on the values chosen for \omega and t. If t is greater than \omega, the integral should give a small number, and if t is close to \omega, the integral should give some non zero value. Any help would be appreciated.