What is the Fourier-series for a function with a random period?

  • Thread starter Niles
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In summary, the student is attempting to find the Fourier-series for the function g(x) with a random period p, using the result from f(x). They make a substitution and solve for the equation, remembering to substitute in the limit as well.
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Niles
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Homework Statement


I have the Fourier-series for the function:

f(x) = x for -Pi < x < Pi.

I know wish to find the Fourier-series for the function:

g(x) = x for -p < x < p,

where p is a random period.

The Attempt at a Solution


Ok, the obvious thing here is to use the result from f(x). If thought that if I define f(p*x/Pi) = g(x) = f(y), then I just insert x=y*Pi/p. But in the result of this exercise, they have an additional p/Pi in front of the sum. Where does this come from?
 
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  • #2
I solved it - just substitute, and then you have a regular equation.. remember to make the substitution in the limit as well!
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It allows us to approximate any periodic function with a combination of simpler trigonometric functions.

2. What is a period in a Fourier series?

The period in a Fourier series is the interval over which the function repeats itself. It can be thought of as the "length" of one cycle of the function. For example, the period of a sine function is 2π, because it repeats itself every 2π units along the x-axis.

3. How are Fourier series used in science and engineering?

Fourier series are used to analyze and approximate signals and functions in various fields such as physics, engineering, and mathematics. They are particularly useful in studying periodic phenomena, such as sound waves, electrical signals, and vibrations.

4. Can any periodic function be represented by a Fourier series?

Yes, any periodic function can be represented by a Fourier series. This is known as the Fourier series expansion theorem, which states that any continuous and periodic function can be approximated by a Fourier series with an infinite number of terms.

5. What is the difference between a Fourier series and a Fourier transform?

A Fourier series represents a periodic function as a sum of sine and cosine functions, while a Fourier transform represents a non-periodic function as a sum of sine and cosine waves with varying frequencies. Fourier transforms are useful for analyzing non-periodic signals, while Fourier series are used for analyzing periodic signals.

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