Proof of Fourier Series Symmetry/Antisymmetry

In summary, the Fourier series for a periodic function that is symmetric or antisymmetric about a point x=a contains only cos(k_{n}(x-a)) or sin(k_{n}(x-a)) terms respectively, with the opposite term dropping out due to symmetry/antisymmetry and the corresponding coefficient being 0. This can be shown using the given integrals and the fact that cosine is symmetric while sine is antisymmetric about a point. The hint suggests setting up an integral centered at the point x=a where the integrand is antisymmetric, resulting in the integral vanishing.
  • #1

Homework Statement


Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only [tex]cos(k_{n}(x-a))[/tex] (including the [tex]a_0[/tex]) or [tex]sin(k_{n}(x-a))[/tex] terms.


Homework Equations


The Fourier expansion for the periodic function, f(x):
[tex]f(x)=a_0 + \sum_{n=1}^{\infty}(a_{n}cos(k_{n}x) + b_{n}sin(k_{n}x))[/tex]

[tex]a_0=\frac{1}{\lambda}\int_{x_0}^{x_0+\lambda}f(x) dx[/tex]


[tex]a_n=\frac{2}{\lambda}\int_{x_0}^{x_0+\lambda}f(x)cos(k_{n}x) dx, n\neq0[/tex]

[tex]b_n=\frac{2}{\lambda}\int_{x_0}^{x_0+\lambda}f(x)sin(k_{n}x) dx, (b_0=0)[/tex]

[tex]k= \frac{2n\pi}{\lambda}\qquad \lambda=period[/tex]

Hint: An integral centered on a point where the integrand is antisymmetric will vanish.

The Attempt at a Solution



I believe I understand what they are saying, but I do not know how to show/prove it. I know that cosine is symmetric about a point, while sine is not. Initially I was thinking that since sine is antisymmetric, for f(x) to be symmetric, the sine terms cannot exist in the expansion (which is exactly what the problem states). I was thinking the same for the opposite case regarding cosines.

The hint leads me to believe I am supposed to set up an integral. However, I do not understand how.

If I am to set up an integral, the best I can figure is that the lower limit should be [tex]a-\lambda[/tex] and the upper limit should be [tex]a+\lambda[/tex]

I think this because for the integral to be centered at point "a", we would need to go 1 full period in both directions.
 
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  • #2
Something just dawned on me. Does anyone think the following is correct?:

If f(x) is symmetric, then b_n is 0 since sine is antisymmetric. Therefore, the sin term in the Fourier series drops out.

If f(x) is antisymmetric, then a_n is 0 since f(x) is antisymmetric and the cosine term in the Fourier series drops out.
 
  • #3
Yup, that's exactly what the hint was getting at.
 
  • #4
Thanks! I can't believe I burned almost 3 hours on that, only to have the answer as soon as I posted it.
 

1. What is the definition of symmetry and antisymmetry in Fourier series?

Symmetry and antisymmetry in Fourier series refer to the behavior of a function under certain transformations. A function is symmetric if it remains unchanged when reflected about a vertical line, while a function is antisymmetric if it changes sign when reflected about a vertical line.

2. How can I determine if a function is symmetric or antisymmetric using its Fourier series?

If a function is symmetric, its Fourier series will only have cosine terms, while an antisymmetric function will only have sine terms. This is because cosine functions are even and retain their value when reflected, while sine functions are odd and change sign when reflected.

3. Can a function be both symmetric and antisymmetric?

No, a function cannot be both symmetric and antisymmetric. If a function is symmetric, it cannot have any odd terms in its Fourier series, and if it is antisymmetric, it cannot have any even terms. Therefore, a function can only exhibit one type of symmetry in its Fourier series.

4. What is the significance of symmetry and antisymmetry in Fourier series?

Symmetry and antisymmetry in Fourier series can simplify the computation and interpretation of the series. By identifying the type of symmetry in a function, we can determine the coefficients of the series and better understand its behavior.

5. How do I use symmetry and antisymmetry to find the Fourier series of a function?

To find the Fourier series of a function using symmetry and antisymmetry, we can use the properties of even and odd functions. If a function is even, we can use the formula for cosine coefficients to find its Fourier series, and if it is odd, we can use the formula for sine coefficients. By considering the symmetry of a function, we can simplify the process of finding its Fourier series.

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