Understanding Fourier Series: Solving Problems and Finding Coefficients

In summary, the conversation is about finding the Fourier series expansion for different functions over specified intervals, specifically for the functions 1 and x, and the relation between even functions and a(n) and b(n). The person asking for help has attempted to find a(0), a(n), and b(n) for the first question, and is unsure about their answers. They are also unclear about what steps to take after finding a(n) and b(n). The other person in the conversation suggests using the fact that f(x) is an even function to determine a(n) and b(n), and then substituting them into the expression for the Fourier series.
  • #1
aobaid
3
0
fourier series pleeeease :(

Hi every one,

I want your help please. I have a difficulty in answering these questions.

Actually it seems simple but I could not answer them because I did not understand the lesson very well.

This homework worths 5% and I have to hand it as soon as possible!



Find the Fourier series expansion for the function over the specified intervals:-

1- f(x)= 1, -∏< x < ∏

2- f(x)= 0, -∏< x < 0 and x^2, 0< x < ∏

Find Fourier cosine series:-

- f(x)= x, 0< x < ∏

Find Fourier sine series:-

- f(x)= -x, 0< x < 1
 
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  • #2
yes, but you have to show attempt to solution and what relations you know of dude. Otherwise you can't get help. It is agaist the forum rules.

If you do this, write the relations you know of, and at least a start from your side. Then I'll kick you in the right direction.
 
  • #3
Thank you for telling me.

Actually I started by finding a(0) and a(n) and b(n)

my attempt for the first question
a(0)= 2 (final answer)

a(n)= I think 0

b(n)= ((-1)^n-(-1)^n)/(n) which is I am really not sure about

And if I get these three answers, should I only substitute it in the main equation OR I have to go in further steps?!

Thank you again my dear
 
  • #4
How did you get 2?

To find a(n) and b(n), note that f(x) is an even function. So what does that tell you about a(n) and b(n)?

Once you get a(n) and b(n) just substitute them into the expression for the Fourier series.
 

What is a Fourier series?

A Fourier series is a mathematical tool used to represent a periodic function as a sum of sines and cosines. It is named after French mathematician Joseph Fourier.

Why are Fourier series important?

Fourier series are important because they can be used to approximate and analyze complex functions, making it easier to solve problems in a variety of fields such as physics, engineering, and signal processing.

What is the formula for a Fourier series?

The formula for a Fourier series is:

f(x) = a0 + ∑n=1 (ancos(nx) + bnsin(nx))

where an and bn are coefficients that determine the amplitude and phase shift of the sine and cosine functions, and n is the frequency of the function.

How do you calculate the coefficients for a Fourier series?

The coefficients for a Fourier series can be calculated using the following formulas:

a0 = (1/π) ∫π f(x) dx
an = (1/π) ∫π f(x)cos(nx) dx
bn = (1/π) ∫π f(x)sin(nx) dx

These integrals can be solved using various techniques, such as integration by parts or trigonometric identities.

What are some applications of Fourier series?

Fourier series have many applications, including in sound and image processing, signal analysis, and solving differential equations. They are also used in fields like music theory, quantum mechanics, and heat transfer.

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