# Problem with the sum of a Fourier series

• Amaelle
In summary, the Fourier sum in the region ##x=2+\epsilon\ ## bounces between ##\log 3 ## and ##\log 2 ##.
Amaelle
Homework Statement
Look at the image
Relevant Equations
Fourier serie
Good day

I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3)
any help would be highly appreciated!

Hi,
Sounds reasonable. However:
what is the sum at ##\ x=2+\epsilon\ ## with ##\ \ 0<\epsilon <<< 1\ ## ?

Amaelle
as I saw it, I can't plug it the function f(x), so I really have no clue ...

The function is periodic, so ##f(2+\epsilon) = f(-{1\over2}+\epsilon)##.
What is ## f(-{1\over2})## ?

Amaelle
f(-1/2)=f(2)=log(3) as they are periodic, right?

And it jumps down to what when you go a little further ?

so maybe I got you
In this region the value of f(-1/2)=0 so the we are in discountinous stepwise function, in which tthe value of f(2+e)
bounces between 2 and 0?

i mean jump between log3 and 0

## f(-{1\over2}) = 0 ## ?

$$\lim_{\epsilon \to 0}f(-\tfrac12 + \epsilon) = \lim_{\epsilon \to 0} |\log(-\tfrac12 + \epsilon)| = \lim_{\epsilon \to 0}\log(\tfrac{2}{1 - 2\epsilon}) = ?$$

BvU said:
## f(-{1\over2}) = 0 ## ?
no I meant f(-1/2 +ε) =log(1+ε) =0 because -1/2 does not belong to the definition domain

pasmith said:
$$\lim_{\epsilon \to 0}f(-\tfrac12 + \epsilon) = \lim_{\epsilon \to 0} |\log(-\tfrac12 + \epsilon)| = \lim_{\epsilon \to 0}\log(\tfrac{2}{1 - 2\epsilon}) = ?$$
log(2)

Amaelle said:
log(2)
but sorry just one question
lim f(-1/2 + epsilon)= lim |log(-1/2+epsilon)|=lim -log(-1/2+ epsilon)=lim log(2/(2epsilon-1)) I think there us a small problem with the sign no?

What is ##\bigl |\log {1\over 2} \bigr |## ?

Amaelle said:
because -1/2 does not belong to the definition domain
Correct, but does it make any difference for the Fourier series ? Do you calculate something different for ##\ (-{1\over 2}, 2]\ ## than for ##\ [-{1\over 2}, 2)\ ## ?

Amaelle
BvU said:
Correct, but does it make any difference for the Fourier series ? Do you calculate something different for ##\ (-{1\over 2}, 2]\ ## than for ##\ [-{1\over 2}, 2)\ ## ?
not does not

BvU said:
What is ##\bigl |\log {1\over 2} \bigr |## ?
|log(2)|=log(2)

Yes; I hope you got it right: ##\ \bigl |\log{1\over 2}\bigr | = |-\log 2\bigr | = \log 2 \ .##

So at ##x=2## the Fourier sum jumps from ##\log 3 ## to ##\log 2 ##. What's the average ?

##\ ##

hutchphd and Amaelle
log(6)/2, thanks a million!

So somehow the subtle difference between ranges ##\ (-{1\over 2}, 2]\ ## and ##\ [-{1\over 2}, 2)\ ## is 'lost in transformation'

##\ ##

Amaelle
BvU said:
So somehow the subtle difference between ranges ##\ (-{1\over 2}, 2]\ ## and ##\ [-{1\over 2}, 2)\ ## is 'lost in transformation'

##\ ##
yes , thanks a million, that was a nice explanation!

## 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 is used to approximate complex functions and is commonly used in signal processing and image analysis.

## 2. What is the problem with the sum of a Fourier series?

The problem with the sum of a Fourier series is that it may not converge to the original function. This is known as the Gibbs phenomenon and results in overshoots and oscillations near discontinuities in the function.

## 3. How can the problem with the sum of a Fourier series be minimized?

The problem with the sum of a Fourier series can be minimized by using a larger number of terms in the series. Additionally, using a different type of series, such as a Chebyshev series, can also help reduce the error.

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

A Fourier series is used to approximate a periodic function, while a Fourier transform is used to decompose a non-periodic function into its frequency components. The Fourier transform is a continuous function, while the Fourier series is a discrete function.

## 5. Are there any real-world applications of Fourier series?

Yes, Fourier series have numerous real-world applications, including signal processing, image analysis, and data compression. They are also used in fields such as physics, engineering, and economics to model and analyze complex systems.

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