Solving Integrals Involving Complex Exponentials

[Xyius]

Hello all,

I am working on a Physics problem and it boils down to solving the following integral.

$$\int_0^{∞}e^{-\frac{t^2}{\tau^2}}e^{i\omega_{kn} t}\cos(\omega_0 t)$$

What I did was re-write the cosine term as..

$$cos(\omega_0 t)=e^{i\omega_0 t}$$

Where in the above expression it is understood that one takes the real part of the exponential. I then plugged it into the above integral expression, completed the square inside the exponential, then used the following identity.

$$\int_{-∞}^{∞}e^{-a(x+b)^2}dx=\sqrt{\frac{\pi}{a}}$$

I then added a factor of 1/2 in the above expression since the limits are only going from 0 to ∞.

But I have a few questions / problems with this.

1. What if the trigonometric function were a sine? My method I did here would still produce the same answer. Is this correct?

2. I have a problem with combining the real part of one complex exponential with the entire part of another complex exponential. Doesn't seem like it would be okay to do.

3. Are there any other ways to integrate this?

Thank you for your time.

 

[LCKurtz]

Hello all,

I am working on a Physics problem and it boils down to solving the following integral.

$$\int_0^{∞}e^{-\frac{t^2}{\tau^2}}e^{i\omega_{kn} t}\cos(\omega_0 t)$$

What I did was re-write the cosine term as..

$$cos(\omega_0 t)=e^{i\omega_0 t}$$

Where in the above expression it is understood that one takes the real part of the exponential. I then plugged it into the above integral expression, completed the square inside the exponential, then used the following identity.

$$\int_{-∞}^{∞}e^{-a(x+b)^2}dx=\sqrt{\frac{\pi}{a}}$$

I then added a factor of 1/2 in the above expression since the limits are only going from 0 to ∞.

But I have a few questions / problems with this.

1. What if the trigonometric function were a sine? My method I did here would still produce the same answer. Is this correct?

2. I have a problem with combining the real part of one complex exponential with the entire part of another complex exponential. Doesn't seem like it would be okay to do.

That step worries me too. Why don't you just substitute$$
\cos(\omega_0t) = \frac{e^{i\omega_0t}+e^{-i\omega_0t}}{2}$$and just work it out? Just a suggestion because I haven't tried it.

 

[Xyius]

Don't know why I didn't think of that! Thanks! I'll go through it now.