Fourier series and even/odd functions

In summary, the conversation discusses the solution to a PDE, which is the sum of u(rho, phi) = [A_n*cos(n*phi)+B_n*sin(n*phi)]*rho^n. The problem is to find the general solution when rho=c, and it is given that u in this point equals sin(phi/2) when phi is between 0 and 2*pi. The question is whether A_n can be discarded since sin(phi/2) is an odd function. The answer is that A_n cannot be discarded because the interval is [0,2pi] and not [-L,L], and therefore the integral of an even function times an odd function will not necessarily vanish.
  • #1
Niles
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[SOLVED] Fourier series and even/odd functions

Homework Statement


I found the solution to a PDE in this thread: https://www.physicsforums.com/showthread.php?t=224902 (not important)

The solution is the sum of u(rho, phi) = [A_n*cos(n*phi)+B_n*sin(n*phi)]*rho^n.

I have to find the general solution, where rho=c, so I equal rho = c, and I am told that u in this point equals sin(phi/2) when phi is between 0 and 2*pi.

I must find the Fourier-coefficients (since it is a Fourier-series).

My questions are:

Since sin(phi/2) is an ODD function, can I discard A_n and just find B_n? That is what I would do, but in the solutions in the back of my book they find A_n as well. Why is that?! The book even says that for an odd function, the Fourier-series only contains sine, so A_n can be discarded, but they still find it. Can you explain to me why A_n must be found as well?
 
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  • #2
The integral of an even function times and odd function will generally vanish only if you are integrating over an interval symmetric around the origin, like [-L,L]. Your interval here is [0,2pi]. The A_n's don't automatically vanish.
 
  • #3
Ahh, I see.

You have helped me very much lately. Thank you.
 

Related to Fourier series and even/odd functions

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions with different amplitudes, frequencies, and phases. It is used to analyze and approximate functions in fields such as signal processing and physics.

2. How is an even function defined?

An even function is a function where f(-x) = f(x) for all values of x. This means that the function is symmetric about the y-axis, and its graph is unchanged when reflected across the y-axis. An example of an even function is f(x) = x^2.

3. How is an odd function defined?

An odd function is a function where f(-x) = -f(x) for all values of x. This means that the function is symmetric about the origin, and its graph is unchanged when rotated 180 degrees around the origin. An example of an odd function is f(x) = x^3.

4. What is the relationship between even and odd functions?

An even function can be written as a sum of two odd functions, and an odd function can be written as a difference of two even functions. Additionally, the product of two even functions is an even function, while the product of an even and an odd function is an odd function.

5. How are Fourier series used to analyze even and odd functions?

For an even function, only cosine terms will be present in the Fourier series, since sine terms would result in a value of 0 due to the symmetry of the function. On the other hand, for an odd function, only sine terms will be present in the Fourier series, since cosine terms would result in a value of 0 due to the symmetry of the function. This simplifies the computation of the Fourier series coefficients for these types of functions.

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