Solving Integral of e^wix for Particle Location

[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.

 

[HallsofIvy]

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

How did you get that last part?
\LARGE {\int_0^t {e^{i\omega t^&#039;}}dt^&#039;} = \frac{e^{i\omega t}}{i\omega}- \frac{1}{i\omega{ = \frac{1}{i\omega}(cos(\omega t)+ i sin(\omega t)+ 1)= \frac{1}{\omega}sin(\omega t)- \frac{1}{\omega}i cos(\omega t)+ \frac{i}{\omega}[/itex]<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 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. </div> </div> </blockquote>