Expanding Fourier Series of $f(x)=\pix$ where $0<x<\pi$ (even and odd)
$a_0=\frac{4}{T}\int_{0}^\frac{T}{2}f(x)=\frac{4}{2\pi}\int_{0}^\pi \pix=\pi x\frac{x^2}{2}_0^\pi=\pi$
$$a_n=\frac{4}{T}\int_{0}^\frac{T}{2}f(x)\cos\frac{2\pi nt}{T} =\frac{4}{2\pi}\int_{0}^\pi (\pix)\cos nx=\frac{2}{\pi}(\frac{\pi}{n}\sin nt\frac{x}{n}\sin nt+\frac{1}{n^2}\cos nt)_0^\pi=\frac{2}{n^2\pi}((1)^n1)$$ so $a_n=\dfrac{4}{\pi n^2}$ for even $n$ and $0$ for odd $n$.
for $\sin$
$$b_n=\frac{4}{T}\int_{0}^\frac{T}{2}f(x)\sin\frac{2\pi nt}{T} =\frac{2}{n}$$
sajjad
I'm the student of BSc. on Tabriz technical collage on "Computer technology engineering". Mainly our mathematical courses are about "Engineering mathematics" (Fourier series , Integration's , so on ) and "Discrete Mathematics" (indirect proof,inference ,so on)
Updated on October 22, 2020Comments

sajjad about 3 years
Please help me solve this Fourier series and correct my solution if it is wrong. it's a nonperiodic function which we need to write its Fourier series (even and odd) :
$ f(x)=\pi  x $ ; $ 0<x<\pi $
I have reached cosine extension(even) as follows:
$ \phi(x)= \begin{cases} \text{$\pix$ ; $0<x<\pi$},\\ \text{$x\pi$ ; $\pi<x<0$} \end{cases} $
My result was $a_{0}=\pi$ and
$ a_{n}= \begin{cases} \text{$\dfrac{4}{\pi n^2}$ ; if $n$ is even}\\\text{0 ; if $n$ is odd} \end{cases} $
and I have found follows for sinus extension(odd) :
$ \phi(x)= \begin{cases} \text{$\pix$ where : $0<x<\pi$},\\ \text{$\pix$ where : $\pi<x<0$} \end{cases} $
$a_{0}=0$ and $b_{n}=\dfrac{2}{n}$ .
I would appreciate if put your solution as "answer".

hardmath about 8 yearsAre you intending to create a discontinuity at $x=0$? I wonder if you've misunderstood the original problem.

sajjad about 8 yearsyes we didn't assume $x$ in all examples of classroom.

Soba noodles about 8 yearsYour first and second $\phi(x)$ are completely different functions with different fourier expansions. Perhaps you meant $\phi(x)=x$ for the part between $\pi$ and 0?

sajjad about 8 yearsThe given equation is not a periodic function so we need to assume that it is but with a little differences which $a_{n}=0$ on $sin$ and $b_{n}=0$ on $cos$.

MPW about 8 yearsDo you intend to have an even or odd extension, with period $2\pi$?

sajjad about 8 yearsof course I mean $cos$ for even and $sin$ for odd

sajjad about 8 yearsI would be appreciate if you share your answer with me (as an "answer" please).


sajjad about 8 yearsCan you please take a look on this one too : math.stackexchange.com/questions/1492740/…