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:

$$\LARGE {\int_0^t {e^{i\omega t^'}}dt^'} = \frac{1}{i\omega}\left(e^{i\omega t}-1\right)$$

Do you see why?

 

[skateboarding]

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

$$\LARGE {\int_0^t {e^{i\omega t^'}}dt^'} = \frac{1}{i\omega}\left(e^{i\omega t}-1\right)$$

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.

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

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.

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

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.