Fourier Series: Simplified Explanation for Beginners

In summary, the function f(x) = e^{-cx} for 0 < x < \pi and f(x) = e^{cx} for -\pi < x < 0 can be written as a cosine series. The Fourier series for even functions only have cosine terms, and since this function is even, the sine terms have zero coefficients. To calculate the coefficients of the cosine terms, we can use integration by parts and evaluate the function at -\pi and \pi. However, in this case, the function is even and the sine terms are odd, so their coefficients will be 0. This means that the function can be simplified to only have cosine terms in its Fourier series.
  • #1
Brewer
212
0
Can someone explain this to me simply. I just plain do not get Fourier Series.

I've got a question that says:
show that
f(x)=exp(-cx) for 0<x<pi
=exp(cx) for -pi<x<0

can be written as a cosine series that's way too complicated for me to work out how to write here.

I have no idea where to start. My lecture notes are of very little help, and all the websites and textbooks seem to be of no help at all.

My only thought of where to start is to write e as functions of cos and sin, but there are no sin's in the answer, so I'm a little dubious about this. If anyone has a way of explaining this to dumbasses, then I'd love to hear it!
 
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  • #2
Any function can be considered the sum of an even function and an odd function. The Fourier series for even functions have only cosine terms, while for odd functions have only sine terms. The function you have is even, so the sine terms have zero coefficients.
 
  • #3
I still struggle to understand what you mean though. How are the coefficients worked out?

Thanks though.
 
  • #4
Almost all "Fourier coefficient" problems can be calculated using "integration by parts".

In this case, where your interval is from [itex]-\pi[/itex] to [itex]\pi[/itex] the coefficient of sin(nx) in the Fourier series for f(x) would be
[tex]\frac{1}{\pi}\int_{-\pi}^\pi f(x)sin(nx)dx[/tex]
the "constant term" (corresponding to cos(0x)) would be
[tex]\frac{1}{2\pi}\int_{-\pi}^\pi f(x) dx[/itex]
and the coefficient of cos(nx) for n> 0
[tex]\frac{1}{\pi}\int_{-\pi}^\pi f(x)cos(nx)dx[/tex]

In the case of your function, the coefficient for sin(nx) is
[tex]\frac{1}{\pi}\int_{-\pi}^0 e^{cx}sin(nx)dx+\frac{1}{\pi}\int_0^\pi e^{-cx}sin(nx)dx[/tex]
Now, in the first let [itex]u= e^{cx}[/itex] and [itex]dv= sin(nx)dx[/itex]. That will give an integral in [itex]e^{cx}cos(nx)[/itex]. Do the same thing and you will get back to [itex]e^{cx}sin(nx)[/itex]. Solve that algebraically for the integral.

However, as mathman pointed out, since this function is even and sine is odd, f(x)sin(nx) is an odd function. It's anti-derivative will be even so evaluating at [itex]-\pi[/itex] and [itex]\pi[/itex] and subtracting gives 0.

Essentially, the Fourier sine terms give the "odd part" of a function and the cosine terms give the "even part". Since this function is even, it has no sine terms. Getting coefficient 0 is exactly right.
 

Related to Fourier Series: Simplified Explanation for Beginners

1. What is a Fourier Series?

A Fourier Series is a mathematical tool used to represent periodic functions as a sum of simpler trigonometric functions. It was developed by Joseph Fourier in the early 19th century and has a wide range of applications in physics, engineering, and other fields.

2. How does a Fourier Series work?

A Fourier Series works by breaking down a periodic function into a series of simpler sine and cosine functions with different frequencies and amplitudes. These simpler functions are then combined to approximate the original function. The more terms included in the series, the closer the approximation will be to the original function.

3. What is the significance of Fourier Series?

The significance of Fourier Series lies in its ability to simplify complex periodic functions into simpler components, making it easier to analyze and understand these functions. It is also used extensively in signal processing, image processing, and other fields to remove noise and extract useful information from data.

4. Can Fourier Series be applied to non-periodic functions?

No, Fourier Series can only be applied to functions that are periodic. However, there are other mathematical tools, such as Fourier Transforms, that can be used to analyze non-periodic functions.

5. Is it necessary to have a deep understanding of mathematics to understand Fourier Series?

While a basic understanding of mathematics is helpful, it is not necessary to have a deep understanding to grasp the concept of Fourier Series. With some patience and practice, beginners can understand the basics of Fourier Series and its applications.

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