Can 1/a Be Expanded Using Fourier Expansion?

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Homework Help Overview

The discussion revolves around the possibility of expanding the term 1/a using Fourier expansion, with participants exploring the nature of Fourier series and their application to constants versus functions.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants question whether 1/a, as a constant, can be treated in the context of Fourier series, with some suggesting that constants have trivial expansions. Others inquire about the expansion of functions like 1/z and f(x) = (1 - x/a), raising concerns about periodicity.

Discussion Status

The discussion is active, with various interpretations being explored regarding the nature of the term 1/a and its applicability to Fourier expansion. Some participants have provided insights into the characteristics of Fourier series, while others are seeking clarification on specific functions.

Contextual Notes

There is an underlying assumption that the function in question must meet certain criteria for Fourier expansion, such as periodicity, which is being discussed among participants.

kaizen.moto
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Dear all,

Iam just wondering whether the term 1/a can be expanded using Fourier expansion. If it does, please let me know how to to do this.

Thank for any kind help.
 
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What do you mean bu "the term 1/a"? Normally, we expand functions in Fourier series, not numbers. Of course, we can think of f(z)= 1/a as the constant function. In that case, the Fourier series is simply (1/a)+ 0 sin(z)+ 0 cos(z)+ ...

If you mean f(z)= 1/z, then, yes, you can find a Fourier series for 1/z using the standard formulas. It will not converge at z= 0, of course.
 


How about f(x) = (1 - x/a), what would be the solution after Fourier expansion?
 
Any constant is a finite Fourier series with all the sine and cosine terms having 0 coeifficients. There is nothing to calculate.
 


kaizen.moto said:
How about f(x) = (1 - x/a), what would be the solution after Fourier expansion?

Your f(x) needs to be periodic to have a FS.
 
Moderator's note: I copied several posts from the thread that was started in the Mathematics technical section.
 

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