- #1

Xyius

- 508

- 4

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

[tex]\int_0^{∞}e^{-\frac{t^2}{\tau^2}}e^{i\omega_{kn} t}\cos(\omega_0 t)[/tex]

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

[tex]cos(\omega_0 t)=e^{i\omega_0 t}[/tex]

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.

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

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.