Why is e^-ipi equal to 1 in complex Fourier series?

[Poop-Loops]

I get how to do them, I just have one question.

An example that my prof. handed out has this:

With the f(x) being 0 from -pi to 0, 1 from 0 to pi/2 and 0 from pi/2 to pi.

But my question is when he has the last line in that picture. He has e^-ipi -> 1? I'm not understanding that step he's doing there. I'm doing the first problem in the complex Fourier series section of the homework and I can't seem to simplify anything, so I'm guessing I'm missing some crucial relation here.

EDIT: Actually, I have no idea what the "n=m" part is either. I don't see "m" anywhere in the book.

 

[matt grime]

n and m are just dummy variables. They don't mean anything, you could just as easily use r, alpha or squiggle to denote elements in the index set.

There's no n=m on that page was there? n=4m, n=4m+2 and n=4m+1 or 3.

You can either sum over even n, or set n=2m and sum over all m.

 

[Poop-Loops]

Yeah I don't get the n=4m n=4m+2 etc parts.

What about the rest? Simplifying e^-ipi as 1?

 

[HallsofIvy]

Yeah I don't get the n=4m n=4m+2 etc parts.

What about the rest? Simplifying e^-ipi as 1?

Yes, that's true.

e^{i\theta}= cos(\theta)+ i sin(\theta)
so
e^{-i\pi}= cos(-\pi)+ i sin(-\pi)= 1
since cos(-\pi)= cos(\pi)= 1 and sin(-\pi)= sin(\pi)= 0.

 

[cliowa]

Yes, that's true.

e^{i\theta}= cos(\theta)+ i sin(\theta)
so
e^{-i\pi}= cos(-\pi)+ i sin(-\pi)= 1
since cos(-\pi)= cos(\pi)= 1 and sin(-\pi)= sin(\pi)= 0.

Unless I'm completely mistaken, that's wrong. It should be
e^{-i\pi}= cos(-\pi)+ i sin(-\pi)= -1
since cos(-\pi)= cos(\pi)= -1 and sin(-\pi)= -sin(\pi)= 0.

 

[Poop-Loops]

Yeah, cos(pi) = -1

But I can't believe I forgot that... Wow... this makes the homework REALLY easy now. Thanks a lot! :D

 

[Poop-Loops]

Just another quick question. Right now I had to do the Fourier series the normal way with An and Bn constants and then do it with Cn constants the complex way.

The former way gave me alternating negative and positive terms and this way gives me either positive or negative terms straight across. Is there a common pitfall or something that I should check, or is it really problem specific? I'm not expecting someone to just say "oh yeah, you did blah blah" or something, but it's always worth a shot. :)

The terms have sines and cosines in them, too. So I just don't know how to make it alternate.