How to Find Fourier Coefficients for a Given Function

[Kqwert]

Homework Statement

Finding fourrier coefficients by observation

Relevant Equations

No eq. posted

Hello,

I need help with question #2 c) from the following link (already LateX-formatted so I save some time...):
https://wiki.math.ntnu.no/_media/tma4135/2017h/tma4135_exo1_us29ngb.pdf

I do understand that the a0 for both expressions must be the same, but what about an and bn? I don't understand how you find them, given that we'll have an*cos(nx)/bn*sin(nx) in the first case while we will have an*cos(3nx)/bn*sin(3nx) in the second case.

 

[BvU]

we'll have an*cos(nx)/bn*sin(nx) in the first case while we will have an*cos(3nx)/bn*sin(3nx) in the second case

I don't see them in your attempt at solution

anyway, posts that show no attempt can't be assisted in: PF rules

 

[Kqwert]

Okay,

so
$$g(x) = ao_{0} + \sum_{n=1}^{inf} ao_{n}cos(nx)+bo_{n}sin(nx)$$
$$f(3x) = a_{0} + \sum_{n=1}^{inf} a_{n}cos(3nx)+b_{n}sin(3nx)$$

and $$g(x) = f(3x)$$

I understand that
$$ao_{0} = a_{0}$$, but not sure what to do with the an and bn parts, to make them equal??

Anyways,

think we have to compare the sin and cos terms?

i.e.

$$ao_{3}cos(3x) = a_{1}cos(3x)$$
so $$ao_{3} = a_{1}$$ ?

 

[BvU]

You make life difficult for yourself:

what to do with the a_n and b_n parts, to make them equal?

You don't want to make the $a_n$ equal at all !
Try using $m$ as summation variable in the second expression and find a relationship between $a_n$ and $a_m$

 

[BvU]

A tip for if you are in a hurry: try a simple example $f$, for instance $f(x) = \cos x$

 

[Kqwert]

I edited post #3. Is that correct?

 

[BvU]

That makes the thread rather difficult to follow, but I think you get the idea, yes.

It's really a very simple exercise if you look 'through' it, isn't it !

 

[BvU]

Hold it !
In your notation, the alternative $ao_{n} = a_{3n}$ looks more sensible

again: check with $f=\cos x$

 

[Kqwert]

Thanks! not sure how to check that though?

 

[BvU]

What is the Fourier series for $\cos x$ ?
Idem $\cos 3x$ ?

 

[Kqwert]

$$cos(x) = \sum_{n = 1}^{inf} a_{n}cos(nx)$$
where every $$a_{n}$$ apart from $$a_{1}$$ is zero..?

Similarly,

we'll have
$$cos(3x) = a_{3}cos(3x)$$

 

[BvU]

In your notation, the alternative $ao_{n} = a_{3n}$ looks more sensible
again: check with $f=\cos x$

The above is of course bogus. As you debunked correctly in #11:
If $f(x) = \cos x \Rightarrow a_1 = 1$ and all other $a_n = 0$ then
$\ \ g(x) = f(3x)$ has $a_3 = 1$ and all other $a_n = 0$
which you can easily generalize.

In other words $\ \ ao_{3n} = a_n \ \ \forall n\ \$ -- as you concluded in the edited post #3.