1. ## 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)

2. 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 $[-\pi,\pi]$

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

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

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

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

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

so we get

$f(t) \approx \frac{1}{2} +\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{\sin[(2n-1)t]}{2n-1}$

3. I have A0 equal to 1/(root(2)). That is still goofing me up.

4. Originally Posted by smellatron
I have A0 equal to 1/(root(2)). That is still goofing me up.
How did you cacluate it?

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

the 1/2 fits very nicely.

6. 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.

7. 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...