Writing Fourier Series for Open and Closed Intervals

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SUMMARY

The discussion clarifies that when writing Fourier series for functions defined on the intervals [0, 2π] and [0, 2π), there is no difference in the resulting series. This is because Fourier series rely on integrals, such as ∫0π f(x)sin(nx)dx, which remain unchanged even if the function differs at a single point. Thus, the continuity or definition of a function at one point does not affect the Fourier series representation.

PREREQUISITES
  • Understanding of Fourier series and their applications
  • Knowledge of integral calculus, specifically integration techniques
  • Familiarity with the concepts of open and closed intervals in mathematics
  • Basic grasp of pointwise convergence and function continuity
NEXT STEPS
  • Study the properties of Fourier series convergence
  • Explore the implications of pointwise and uniform convergence in Fourier analysis
  • Learn about the Dirichlet conditions for Fourier series
  • Investigate the impact of discontinuities on Fourier series representation
USEFUL FOR

Mathematics students, educators in advanced calculus, and anyone studying Fourier analysis or signal processing will benefit from this discussion.

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In a rigorous mathematical course I am talking, it seems to make a difference when I am given a function f and need to write its Fourier series, whether it is defined on [0,2∏] or [0,2∏). What difference does it make for my series whether it is an open or a closed interval?
 
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It doesn't make a difference. If two function differ in one point only (or if one function is not defined in one point), then the Fourier series will still be the same thing. The reason for this is that Fourier series only depend on integrals such as

\int_0^\pi f(x)\sin(nx)dx

But if ##f## changes in one point, then the integral will of course remain the same thing.
 

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