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Solving Integrals Involving Complex Exponentials
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[QUOTE="Xyius, post: 4702971, member: 226245"] Hello all, 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. [B]1. [/B]What if the trigonometric function were a sine? My method I did here would still produce the same answer. Is this correct? [B]2.[/B] 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. [B]3. [/B]Are there any other ways to integrate this? Thank you for your time. [/QUOTE]
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Solving Integrals Involving Complex Exponentials
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