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

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In summary, the given function, f(x) = cos((2*pi*n)/L * x), has a period of L for any value of n. This can be seen through the literature's explanation and the fact that f(x + L) = f(x) for any n. However, considering a specific value of n, such as n = 3, both L and L/n can satisfy the condition (#) f(x + T) = f(x), but only L is the minimal number for which this condition holds. Therefore, the period of f(x) is L/n. This applies to any function of the form cos(kx), which has a period of 2pi/k.
  • #1
wormhole
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[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??
 
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  • #2
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
 
  • #3
actually there is a summation over n
it is just one term from Fourier series...
 
  • #4
So, you need to know why

[tex]\sum_n b_n\cos(2\pi nx/L)[/tex]

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
 
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  • #5
you might have misread the literature
you are right...:smile: (too much reading)

thank you very much
 
  • #6
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?
 
  • #7
cos(x) has period 2pi

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

that enough?
 
  • #8
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?
 
  • #9
yep that looks about right. wolfram's mathworld is your friend for these things
 
  • #10
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)
 
  • #11
thanks guys for your help:smile:
no more question...
 
  • #12
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.
 

1. What is Fourier analysis?

Fourier analysis is a mathematical technique used to decompose a function into its constituent frequencies. It is based on the idea that any complex signal can be represented as a sum of simple sine and cosine waves with different frequencies, amplitudes, and phases.

2. What is a function’s period?

The period of a function is the length of one complete cycle of the function. It is the smallest value of t for which the function repeats itself. In other words, it is the distance between two consecutive peaks or troughs of the function.

3. How is a function’s period related to n in Fourier analysis?

The period of a function is inversely proportional to the value of n, where n is the frequency of the sine or cosine wave in the Fourier series. As n increases, the frequency of the wave increases, resulting in a shorter period for the function.

4. Can a function have multiple periods?

Yes, a function can have multiple periods if it is not a sine or cosine function. In such cases, the function may have different periods for different parts of its graph, depending on the frequency of the underlying waves.

5. How is Fourier analysis used in real-world applications?

Fourier analysis has a wide range of applications in various fields such as signal processing, image and sound compression, and data analysis. It is also used in engineering, physics, and mathematics to study different phenomena and solve complex problems.

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