The value of a Fourier series at a jump point (discontinuity)

In summary, the conversation discusses a function with a jump at x=e and the challenge of dealing with a limit of -infinity and a value of 1 at x=e. It is noted that the function does not satisfy the Dirichlet conditions, making the usual theorem about convergence inapplicable.
  • #1
Amaelle
310
54
Homework Statement
calculate the value of the fourier serie at x=e for the e periodic function (0,e]
f(x)=log(x)
Relevant Equations
Fourier serie
Greetings
according to the function we can see that there is a jump at x=e and I know that the value of the function at x=e should be the average between the value of f(x) at this points
my problem is the following
the limit of f(x) at x=e is -infinity and f(e)=1
how can we deal with such situations?

thank you!
 
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  • #2
Since ##f(0^+) = -\infty##, your function doesn't satisfy the Dirichlet conditions, so the usual theorem about convergence doesn't apply.
 
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Likes Amaelle
  • #3
LCKurtz said:
Since ##f(0^+) = -\infty##, your function doesn't satisfy the Dirichlet conditions, so the usual theorem about convergence doesn't apply.
thanks a million!
 

Related to The value of a Fourier series at a jump point (discontinuity)

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is used to approximate an arbitrary periodic function with a combination of simpler trigonometric functions.

2. What is a jump point (discontinuity)?

A jump point or discontinuity is a point on a function where there is a sudden change or jump in the value of the function. This can occur due to a sudden change in the input or due to a sudden change in the behavior of the function itself.

3. Why is the value of a Fourier series at a jump point important?

The value of a Fourier series at a jump point is important because it determines the accuracy of the approximation of the original function. If the jump point is not taken into account, the Fourier series may not accurately represent the behavior of the original function at that point.

4. How is the value of a Fourier series at a jump point calculated?

The value of a Fourier series at a jump point can be calculated using the jump conditions, which take into account the value of the function and its derivatives at the jump point. These conditions are used to adjust the coefficients of the Fourier series to accurately represent the behavior of the function at the jump point.

5. Can a Fourier series accurately represent a function with a jump point?

Yes, a Fourier series can accurately represent a function with a jump point as long as the jump conditions are taken into account. However, the accuracy of the representation may depend on the number of terms in the series and the smoothness of the function at the jump point.

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