Finding a fourier series for a cosine function

[Susanne217]

Hi I have been working on a example and have worked it out as this

Homework Statement

f(x) = \( \left( \left \begin{array}{ccc}0 & \mathrm{for} & -\pi < x \leq 0 \\ cos(x) & \mathrm{for} & 0 < x < \pi\end{array}

where f is defined on the interval ]-\pi,\pi[.

Find the corresponding Fourier series for f.

and writ the sum for x = p \cdot \pi where p \in \mathbb{Z}

Homework Equations

Complex Fourier series is defined as

\sum_{n=-\infty}^{\infty} c_{n} \cdot e^{i\cdot n \cdot t}

where c_{n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) \cdot e^{-i \cdot n \cdot t} dt

The Attempt at a Solution

First I find c_{0} = \frac{1}{2\pi} \int_{-\pi}^{\pi} cos(t) dt = [\mathrm{sin(t)}]_{\mathrm{t} = -\pi}^{\pi} = 0

Next I find

c_{n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} (cos(t) \cdot cos(n \cdot t) - cos(t) \cdot sin(n \cdot t) \cdot i) dt

c_{n} = [\frac{{sin(n-1)\cdot t}}{2 \cdot (n-1)} + \frac{{sin(n11)\cdot t}}{2 \cdot (n+1)}-(\frac{{cos(n-1)\cdot t}}{2 \cdot (n-1)} - \frac{{cos(n+1)\cdot t}}{2 \cdot (n+1)}) \cdot i]_{t=-\pi}^{\pi} = \frac{sin(n+1)\cdot \pi}{2\cdot(n+1)}

which I then multiply by \frac{1}{2\pi} from which I get

c_{n} = \frac{sin(n+1)}{2(n+1)}

Where I end up with the Fourier series

\sum_{n=-\infty}^{\infty} \frac{sin(n+1)}{2(n+1)} \cdot e^{i \cdot n \cdot x}

Have I got it right? I fairly new to Fourier series so I would be very happy if someone would reflect on my work and comment it please.

p.s. Not sure about question two. Do I insert the value of into the sum?

Thank you

Best Regards

Susanne

 

[latentcorpse]

remember that c_0=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) dt

but f(t) is not constnat on the interval [-\pi,\pi]...

and so c_0=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) dt=\frac{1}{2 \pi} \int_{-\pi}^0 0 dt + \frac{1}{2 \pi} \int_0^{\pi} \cos{t} dt=\frac{1}{2 \pi} \int_0^{\pi} \cos{t} dt=\frac{1}{2 \pi} \sin{t} |_{0}^{\pi}=0 which is what you got.

but you'll have to do the same when calculating c_n which will probably change things.

 

[Susanne217]

remember that c_0=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) dt

but f(t) is not constnat on the interval [-\pi,\pi]...

and so c_0=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) dt=\frac{1}{2 \pi} \int_{-\pi}^0 0 dt + \frac{1}{2 \pi} \int_0^{\pi} \cos{t} dt=\frac{1}{2 \pi} \int_0^{\pi} \cos{t} dt=\frac{1}{2 \pi} \sin{t} |_{0}^{\pi}=0 which is what you got.

but you'll have to do the same when calculating c_n which will probably change things.

Hi

following your line of thought I then get

c_{n} = \frac{1}{2\pi} \int_{\pi}^{pi} cos(t) \cdot cos(n \cdot t) - (cos(t) \cdot sin(n \cdot t) \cdot i) dt = \frac{1}{2\pi} \int_{\pi}^{0} cos(t) \cdot cos(n \cdot t) - (cos(t) \cdot sin(n \cdot t) \cdot i) dt + \frac{1}{2\pi} \int_{0}^{pi} cos(t) \cdot cos(n \cdot t) - (cos(t) \cdot sin(n \cdot t) \cdot i) dt

where

c_{n} = (\frac{-1}{2(n+1)\cdot \pi}-\frac{1}{2(n-1)\cdot \pi}) \cdot sin(n \cdot \pi)

does that look better?

Best Regards
Susanne

 

[latentcorpse]

c_n=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) e^{-int} dt=\frac{1}{2 \pi} \int_{-\pi}^{0} 0 dt + \frac{1}{2 \pi} \int_{0}^\pi} \cos{t} e^{-int} dt

do you understand why that is?

so c_n=\frac{1}{2 \pi} \int_{0}^\pi} \cos{t} e^{-int} dt=\frac{1}{2 \pi} \int_{0}^\pi} \cos{t} \( \cos{nt}-i \sin{nt} \)
as the first term disappears entirely

 

[Susanne217]

c_n=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) e^{-int} dt=\frac{1}{2 \pi} \int_{-\pi}^{0} 0 dt + \frac{1}{2 \pi} \int_{0}^\pi} \cos{t} e^{-int} dt

do you understand why that is?

so c_n=\frac{1}{2 \pi} \int_{0}^\pi} \cos{t} e^{-int} dt=\frac{1}{2 \pi} \int_{0}^\pi} \cos{t} \( \cos{nt}-i \sin{nt} \)
as the first term disappears entirely

Hi again yes the reason why that term disappears it do to basic calculus.

I have tried the new version of c_{n}

and I get

c_n = (\frac{-1}{4(n+1)\cdot \pi} - \frac{1}{4(n-1)\cdot \pi}) \cdot sin(n\pi) + \frac{cos(n\cdot \pi) -1}{2 \cdot n \cdot \pi} \cdot i

Hopefully this looks better?

Best Regards
Susanne

 

[latentcorpse]

hmmm.
first of all i meant to put a bracket in my last line:

c_n=\frac{1}{2 \pi} \int_0^{\pi} \cos{t} (\cos{nt}-i \sin{nt}) dt

then it's just integrating by parts.

i'll assume you managed that ok.

you have c_n=(\frac{-1}{4(n+1) \cdot \pi}-\frac{1}{4(n-1) \cdot \pi}) \cdot \sin{(n \pi)} + \frac{\cos{(n \cdot \pi)}-1}{2 \cdot n \cdot \pi} \cdot i

but \sin{n \pi}=0 and \cos{n \pi}=(-1)^n which will simplify things i hope...