# Exploring the Period & Fourier Series of f(x)

• hbomb
In summary: Even and odd are simply the terms used to describe whether or not f(x) is a multiple of -1 and 1 respectively.
hbomb
Let f(x) = 0, -2< x <0 and x, 0< x <2
f(x) repeats this pattern for all x

a) What is the period of f(x)?
b) Is f(x) even, odd, or neither?
c) Find the Fourier Series for f(x).

a) I found that the period is 2
b) odd
c) I'm not even sure I got close so I'm not even going to bother putting it up

a) Do you know the definition of periodicity? A function f(x) is periodic of period L>0 if for all x we have f(x)=f(x+L). When we talk about the period of a function, we usually talk about the smallest such number L.

So according to your answer for a), the period is 2. But for instance, f(-1)=0 but f(-1+2)=1. So 2 isn't the period according to the definition above.

The definition is kind of abstract comparatively to how easy it is to find the period of most functions by just looking at their graph. By looking at the graph, the period is the length of the smallest interval such that the rest of the function is just a repetition(a "copy/paste") of the function in that interval.

b) Again, do you know what the definition of even and odd is or you just flipped a coin? f(x) is even if f(x)=f(-x) for all x. f(x) is odd if f(-x)=-f(x) for all x. So for instance, f(1)=1, but f(-1)=0. So f is not odd.

c) This is just a matter of calculating the two integrals for the coefficients and substituting the answers in the general form of the fouriers series.

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Well, I thought I knew what I was doing. My professor gave us a take home quiz that is due after Thanksgiving and he hasn't showed us how to do piecewise Fourier series. Possibly Monday he'll show us. I figure I could get a head start and learn how to do these problems.

There is no difference btw piecewise and non-piecewise Fourier epansion. Both are simply about calculating the integrals giving the Fourier coefficients.

Take notice of the changes I made in post #2 concerning the definition of even and odd. I had written that f is even if f(x)=f(x) and odd if f(x)=f(-x) which is completely off.

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## 1. What is the Period & Fourier Series of f(x)?

The Period & Fourier Series of f(x) is a mathematical representation of a periodic function using a combination of trigonometric functions. It is used to decompose a periodic function into a sum of simpler trigonometric functions, making it easier to analyze and understand.

## 2. How is the Period & Fourier Series calculated?

The Period & Fourier Series is calculated using a formula known as the Fourier Series formula. This formula involves finding the coefficients of the trigonometric functions that best fit the given periodic function. These coefficients are then used to construct the Period & Fourier Series of the function.

## 3. What is the significance of the Period & Fourier Series in mathematics?

The Period & Fourier Series has a wide range of applications in mathematics, including signal processing, heat transfer, and harmonic analysis. It also plays a crucial role in understanding the behavior of periodic functions and their properties.

## 4. Can the Period & Fourier Series be used for non-periodic functions?

No, the Period & Fourier Series is only applicable to periodic functions. However, there are other methods, such as the Laplace transform, that can be used to represent non-periodic functions using a series of simpler functions.

## 5. How has the Period & Fourier Series impacted other fields besides mathematics?

The Period & Fourier Series has had a significant impact on various fields, such as physics, engineering, and economics. It has allowed scientists and engineers to analyze and understand complex periodic phenomena, making it a valuable tool in problem-solving and research.

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