# Fourier Coefficients

• Mar 30th 2009, 06:21 PM
smellatron
Fourier Coefficients
Find the Fourier coefficients of the piecewise continuousfunction

f(t) = {0 if t <= (less than or equal to) 0 or 1 if t > 0.}

I get why Ck is equal to 0. But why is Bk = 2/(k(pi)) when k is odd. And why is A0 = 1/(root(2)). Thanks for any help.

(sorry its written in straight text)
• Mar 30th 2009, 06:44 PM
TheEmptySet
Quote:

Originally Posted by smellatron
Find the Fourier coefficients of the piecewise continuousfunction

f(t) = 0 if t <= (less than or equal to) 0
1 if t > 0.

I get why Ck is equal to 0. But why is Bk = 2/(k(pi)) when k is odd. And why is A0 = 1/(root(2)). Thanks for any help.

(sorry its written in straight text)

You didn't specify an interval, but I'm guessing you want it on $\displaystyle [-\pi,\pi]$

By definition $\displaystyle a_0=\frac{1}{\pi} \int_{-\pi}^{\pi}f(t)dt = \frac{1}{\pi} \int_{-\pi}^{0} 0 dt + \frac{1}{\pi} \int_{0}^{\pi}1dt =1$

$\displaystyle a_n=\frac{1}{\pi}\int_{0}^{\pi}\cos(nt)dt=\frac{\s in(nt)}{n}\bigg|_{0}^{\pi}=\frac{\sin(\pi n)}{n}=0$

$\displaystyle b_n=\frac{1}{\pi}\int_{0}^{\pi}\sin(nt)dt=-\frac{\cos(nt)}{n}\bigg|_{0}^{\pi}=-\frac{1}{\pi n}(\cos(\pi n)-1)=-\frac{1}{\pi n}[(-1)^n-1]$

note that when ever n is even $\displaystyle b_n=0$ so it can be rewritten as follows

$\displaystyle b_n=\frac{2}{(2n-1) \pi }$ for n=1,2,3...

so we get

$\displaystyle f(t) \approx \frac{1}{2} +\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{\sin[(2n-1)t]}{2n-1}$
• Mar 31st 2009, 12:55 PM
smellatron
I have A0 equal to 1/(root(2)). That is still goofing me up.
• Mar 31st 2009, 02:34 PM
TheEmptySet
Quote:

Originally Posted by smellatron
I have A0 equal to 1/(root(2)). That is still goofing me up.

How did you cacluate it?
• Mar 31st 2009, 05:07 PM
TheEmptySet
Attachment 10732

Here is an immage. $\displaystyle \frac{1}{\sqrt{2}}$ version is in blue and does not fit the function described.

the 1/2 fits very nicely.
• Apr 1st 2009, 04:47 AM
smellatron
That is the answer in the back of the book. I was just confused on how they came up with A0. Are you saying that the answer in the book is possibly wrong? Thanks you for your help.
• Apr 1st 2009, 06:28 AM
TheEmptySet
Quote:

Originally Posted by smellatron
That is the answer in the back of the book. I was just confused on how they came up with A0. Are you saying that the answer in the book is possibly wrong? Thanks you for your help.

I disagree with the back of your book, but you still didn't answer my question how did you calculate it. When you try to calclucate it what do you get...