I {Edit-Solved} Confirmation requested on deriving functions from graphs

AI Thread Summary
The discussion revolves around deriving a function from a periodic graph, initially interpreted as periodic with a 2π interval. The derived function is expressed as y = (a/π)x for specific intervals and zero otherwise. There is confusion regarding how to calculate the Fourier Series (FS) for this function, particularly concerning the integration intervals. It is clarified that the function can be expressed in the standard Fourier Series form, allowing for the calculation of coefficients a_k and b_k. The conversation concludes with the user expressing gratitude for the clarification received.
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So I basically saw this graph specified as a particular waveform in my book while reading Fourier Series. I decided to try and derive its function since once I do that I can easily find the FS. Please find the photo and my attempt below, just need a small confirmation if I'm right/wrong.
So I thought that the graph tries to tell us that the function is periodic after 2π interval. So I tried to derive its function from the graph as follows using the point slope equation form for the points (0,0) & (a,π): ##y= ({a}/{π})*x##

I hope this function is alright and I just need to find its Fourier Series
 

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Actually the graph shows
##y=\frac{a}{\pi}(x-2n\pi)## for ##0<(x-2n\pi)<\pi## for integer n
y=0 for others.
 
anuttarasammyak said:
Actually the graph shows
##y=\frac{a}{\pi}x## for ##2n\pi<x<(2n+1)\pi## for integer n
y=0 for others.
Oh I see. Now I realize how you have represented it an even general manner. Although I am a bit confused now, how should I find FS for the same, I mean I'm confused about the intervalbin which integration for FS will be carried out..
 
Because the function, say f(x), has period of 2##\pi##, it is expressed as
f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty (b_k \sin kx + a_k \cos kx)
You may calculate ##a_k## and ##b_k## in a usual way.
 
anuttarasammyak said:
Because the function, say f(x), has period of 2##\pi##, it is expressed as
f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty (b_k \sin kx + a_k \cos kx)
You may calculate ##a_k## and ##b_k## in a usual way.
Thank you so much! Now I get it🙏🏻
 
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