# Complex Fourier series has a singular term

1. Apr 5, 2009

### Ragnord

1. The problem statement, all variables and given/known data

Find the complex Fourier series for f(t)=t(1-t), 0<t<1

2. Relevant equations

$$\sum_{n=-\infty}^{\infty}c_{n}e^{2in\pi t}$$

where $$c_{n}=\int_{0}^{1}f(t)e^{-2in\pi t}dt$$

3. The attempt at a solution

I've worked out that c$$_{n}=-1/(2n^2 \pi^2)$$. The problem is that for n=0, it is singular. Is there some way around this or does it mean that the complex Fourier series doesn't exist?
I tried using maple to graph the series with the n=0 term omitted and it comes out to the right shape, but is shifted vertically down some, leading me to believe that the singular term should be replaced by a constant or something.

2. Apr 6, 2009

### Billy Bob

$$c_0=\int_0^1 f(t)\,dt$$

3. Apr 6, 2009

### Ragnord

Well that was easy, just like a real Fourier series. Thanks.
I'm interested in knowing why that's the case though, I haven't seen anything about doing anything special for $$c_{0}$$ in anything I've seen about complex Fourier series.

4. Apr 7, 2009

### Billy Bob

Just plug in n=0 before integrating instead of after!