Fourier Analysis: Function's Period & Its Relation to n

[wormhole]

[more on the same] function's period

i have this function(taken from Fourier analysis):

n - 1,2,3...

f(x) = cos( (2*pi*n)/L * x )

the literature says this function has L period (n parameter):

f(x + L) = cos( (2*pi*n)/L * (x + L) ) =
= cos( 2*pi*n/L * x + 2*pi*n) = cos( 2*pi*n/L * x)

so it's true that L f's period for any n...
but L/n is also f's period because:

f(x + L/n) = cos( (2*pi*n)/L * (x + L/n) ) =
= cos( 2*pi*n/L * x + 2*pi) = cos( 2*pi*n/L * x)


so what happens here??

 

[matt grime]

n varies, and as there is no summation there, one only conclude that when you treat f as unique, rather than a family of functions, one for each n, that you might have misread the literature

 

[wormhole]

actually there is a summation over n
it is just one term from Fourier series...

 

[matt grime]

So, you need to know why

$$\sum_n b_n\cos(2\pi nx/L)$$

has period L and not L/n? Erm, is because n varies not an acceptable anwer?

It doesnt't matter that each individual term may have period less than L, only that the sum has period L.

And assuming b_n are such that that sum makes sense obviously

 

[wormhole]

you might have misread the literature

you are right... (too much reading)

thank you very much

 

[wormhole]

i want to rephrase what i asked before

lets take this function,

f(x)=cos(2*pi*3/L*x)

n is parameter and let's say n=3 so f(x) becomes

f(x)=cos(2*pi*3/L*x)

as i said earlier both numbers L/3 and L
are sutisfying the following condition

(#) f(x + T)=f(x): T f's period

or to be more specific

f(x + L)=f(x) and f(x + L/3)=f(x)

I asked how it can be that both equilities are true...
Meanwhile it occurred to me that this condition (#) alone doesn't says
that T is a period and if we want T to be a period we must demand
that T is also minimal number for which (#) holds

so the answer to my original question is that f's period is L/n

Is what i wrote is correct?

 

[matt grime]

cos(x) has period 2pi

cos(kx) has period 2pi/k for every k in R

that enough?

 

[wormhole]

cos(kx) has period 2pi/k

you mean that cos(kx) has a period 2*p/k?

but what about what i said before that?
...that period is defined by:

1) f(x+L)=f(x)
2) L is minimal among all other numbers

is that correct?

 

[matt grime]

yep that looks about right. wolfram's mathworld is your friend for these things

 

[Organic]

Hi wormhole,

Let us say that L or L/n are circle's radius.

So, in both cases you have f(x+circle)=f(x)

 

[wormhole]

thanks guys for your help
no more question...

 

[matt grime]

Originally posted by Organic
Hi wormhole,

Let us say that L or L/n are circle's radius.

So, in both cases you have f(x+circle)=f(x)

Erm, we could say that, but it would be wrong.