Fourier Series for f on the Interval [-π, π) | Homework Statement

In summary, the Fourier series of ##f## on the interval ##[−π, π)## is given by ##\frac{a_0}{2} + \sum_{n=0}^\infty a_n cos(nx) + \sum_{n=0}^\infty b_n sin(nx)## where ##a_0 = 1, a_n = 0, b_n = \frac{4}{πn} - \frac{4}{πn}(-1)^n##.
  • #1
Calu
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0

Homework Statement


Define ##f : [−π, π) → \mathbb R ## by
##f(x)## = ##−1## if ##− π ≤ x < 0##, ##1## if ##0 ≤ x < π.##
Show that the Fourier series of f is given by
##\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x##

Homework Equations



The Fourier series for ##f## on the interval ##[−π, π)## is given by:

##\frac{a0}{2} + \sum_{n=0}^\infty ancos(nx) + \sum_{n=0}^\infty bnsin(nx)##
(Not quite sure why the LaTex isn't working here, I'm new at this.)
Where a0/2, an, bn are the Fourier coefficients of ##f##.

The Attempt at a Solution



I have attempted to find the Fourier coefficients, however I don't think that they're correct.

I have found
a0/2 = 1
an = 0.
bn = ##\frac {2}{πn} -2\frac {(-1)^n}{n}##

Could somebody tell me if these are correct? If not, I'll post up how I reached those answers.
 
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  • #2
I think you lost a ##\pi## in the second term of your expression for ##b_n##.
 
  • #3
##a_0## isn't correct either.
 

What is a Fourier Series Problem?

A Fourier Series Problem is a mathematical concept that involves representing a periodic function as a sum of simple sine and cosine functions. It is used to solve various problems in mathematics, physics, and engineering.

How do you find the coefficients of a Fourier Series?

The coefficients of a Fourier Series can be found by using the Fourier Series formula, which involves integrating the given function with respect to the period of the function. This process is known as Fourier Analysis.

What are the applications of Fourier Series?

Fourier Series has many applications in different fields such as signal processing, image processing, and data compression. It is also used in solving partial differential equations, which have applications in physics and engineering.

What is the difference between Fourier Series and Fourier Transform?

The main difference between Fourier Series and Fourier Transform is that Fourier Series is used for periodic functions while Fourier Transform is used for non-periodic functions. Fourier Transform also provides frequency and amplitude information, whereas Fourier Series only gives amplitude information.

What are the limitations of Fourier Series?

Fourier Series has some limitations, such as it can only be applied to periodic functions, and the function must be continuous and have a finite number of discontinuities. It also has difficulties in converging for certain functions, and it may not accurately represent functions with sharp corners or edges.

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