# Thread: Fourier Series/Dirichlet Conditions

1. ## Fourier Series/Dirichlet Conditions

Dirichlet conditions

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1. Ok, after plotting/sketching the curves, I think f(x) doesnt violate any dirichlet conditions, and g(x) does. But im not sure which condition g(x) violates?

2. How would i represent f(x) a fourier series? integrate 1/x?

2. Originally Posted by gomes
Dirichlet conditions

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1. Ok, after plotting/sketching the curves, I think f(x) doesnt violate any dirichlet conditions, and g(x) does. But im not sure which condition g(x) violates?

2. How would i represent f(x) a fourier series? integrate 1/x?
You need the periodic extensions of $\displaystyle $$f and \displaystyle$$g$ , and you need the conditions for a fundamental period of $\displaystyle 1$ not $\displaystyle 2\pi$.

Given that, here you should be looking at the absolute integrability condition.

CB

3. thanks, lets say the conditions instead of 2pi or pi etc..are replaced by 1?

4. Originally Posted by gomes
thanks, lets say the conditions instead of 2pi or pi etc..are replaced by 1?
Which of:

$\displaystyle \displaystyle \int_0^1 |f(x)|\; dx <\infty$

and

$\displaystyle \displaystyle \int_0^1 |g(x)|\; dx <\infty$

are true?

CB

5. Thanks! the integral of g(x) is true, but the integral of f(x) is false.

So that means g(x) can be represented as a fourier series and f(x) cant.

Question: So how do i show with calculations that g(x) can be represented as fourier series and f(x) cant? do i just use the formulae below (and use the limits 0 and 1, instead of pi)?

6. Originally Posted by gomes
Thanks! the integral of g(x) is true, but the integral of f(x) is false.

So that means g(x) can be represented as a fourier series and f(x) cant.

Question: So how do i show with calculations that g(x) can be represented as fourier series and f(x) cant? do i just use the formulae below (and use the limits 0 and 1, instead of pi)?

When you adjust these to have the correct fundamental yes, but note (in this case it is obvious as both $\displaystyle $$f and \displaystyle$$ g$ are positive functions, but would be true in general if the absolute integrability condition fails) $\displaystyle $$a_0 only exists for \displaystyle$$ g(x)$

CB