# Fourier Series/Dirichlet Conditions

• Oct 7th 2010, 08:58 AM
gomes
Fourier Series/Dirichlet Conditions
Dirichlet conditions
http://img26.imageshack.us/img26/2387/123zd.jpg

-------------------------------------
http://img829.imageshack.us/img829/1364/12345co.jpg
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?
• Oct 7th 2010, 03:45 PM
CaptainBlack
Quote:

Originally Posted by gomes
Dirichlet conditions
http://img26.imageshack.us/img26/2387/123zd.jpg

-------------------------------------
http://img829.imageshack.us/img829/1364/12345co.jpg
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 $f$ and $g$ , and you need the conditions for a fundamental period of $1$ not $2\pi$.

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

CB
• Oct 8th 2010, 07:32 AM
gomes
thanks, lets say the conditions instead of 2pi or pi etc..are replaced by 1?
• Oct 8th 2010, 07:54 AM
CaptainBlack
Quote:

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

Which of:

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

and

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

are true?

CB
• Oct 8th 2010, 09:32 AM
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)?

http://img69.imageshack.us/img69/6758/123123123nx.jpg
• Oct 8th 2010, 10:17 AM
CaptainBlack
Quote:

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)?

http://img69.imageshack.us/img69/6758/123123123nx.jpg

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

CB