[Q] integral issue

[buduza]

I was recently challenged with solving the integral of e^(i*w*t)dt from negative infinity to infinity, where w is a constant (angular frequency in context, but not important for integration purposes). Any ideas on this would be very helpful.

Also, as I was thinking about the problem I realized that logarithms with negative inputs give outputs in the complex plane [e^(i*pi)=cos(pi)+i*sin(pi)=-1 => ln(-1)=i*pi]. Is there an easy explanation to this?

 

Thanks!

 

-Noah

[sanctus]

To the second part of your question it is just the definition of the complex logarithm:

[math] log(z)=\ln\vert z \vert + i \cdot arg(z)[/math]

see Logarithm - Wikipedia, the free encyclopedia better the short explanation (the link I sent you) than the main article if you are just curious.

 

About your integral, on the, without really thinking, I wonder, if it really is integrable because it turns in circles in the complex plane. But it is very likely that I am saying BS, so do you know if a solution exists?

[buduza]

See, that's what I thought (about it circling around the time axis in complex space). Professor claims that if anyone can give a solid proof of the answer they would automatically get an A in the course and that it took him two days to solve...

 

Thanks for the link by the way.

 

-noah