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Fourier series expansion problem

  1. Aug 11, 2015 #1
    < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >

    hi I've got a problem that I've partially worked but don't understand the next part/have made a mistake?

    f(x)=0 for -π<x<0 and f(x)=x for 0≤x≤π
    i got a0=π/4 and an=0 and bn=0 if n is even and 2/n if n is odd

    so.... f(x)= π/4 + ∑ (2/(2n+1)) * sin(2n+1) * π * x
    n=0
    now i need to show π/4 = 1 - 1/3 + 1/5 ......... using suitable values of x
    how would i go about doing this?
    any help is much appreciated
     
    Last edited by a moderator: Aug 11, 2015
  2. jcsd
  3. Aug 11, 2015 #2

    BvU

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    Hello bbq, welcome to PF :smile: !

    Can you show how you found these Fourier coefficients ?

    Your choices for x are pretty limited, right ? If you try ##x=\pi## (not the right answer I suppose) what does the series give you ?
     
  4. Aug 11, 2015 #3
    http://imageshack.com/a/img911/5796/jhLmo0.jpg [Broken]
    http://imageshack.com/a/img673/1972/xQeDHn.jpg [Broken]
    x=pi gives too high so i must have made a mistake on calculation of ceoffiscients but i don't see it
     
    Last edited by a moderator: May 7, 2017
  5. Aug 11, 2015 #4

    BvU

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    Wow. So you write it as a cosine series ? No sines ? Wouldn't that (##b_n = 0## the coefficents for sin nx for all n) give an even function as a result ?

    [edit] oh, wait, the picture continues further down. Need some time to decrypt...


    And $$a_n \equiv {1\over 2\pi} \int_0^\pi x\cos (nx)\; dx\quad {\rm ?} $$
     
  6. Aug 11, 2015 #5

    BvU

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    I can see ##a_n = 0## for n = even. What if n is odd ?
     
  7. Aug 11, 2015 #6

    BvU

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    could that be f(x)= π/4 + 1/(2π) ∑ (2π/(2n+1)) * sin((2n+1) * x) ? And then you choose x = π/2 to get the (-1)n ?

    (a choice that gets rid of the cosines too ! )

    Modulo further errors on my part ....
     
  8. Aug 11, 2015 #7
    thanks BvU.
    I looked over it and noticed some mistakes, so.. an=( (-1)^n -1) / (π * n^2)
    and bn= - ( (-1)^n ) / n
    i see the π/2 getting rid of cosine but how do i achieve the π/4 = 1 - 1/3 + 1/5 ......... using suitable values of x.
    do i set x=π/4 then get 1 for ??
     
  9. Aug 11, 2015 #8

    BvU

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    You're real close now !
    But I suspect the (-1)n-1 and (-1)n is wishful thinking ? Could you explain ?

    On the one hand you have ##f(x) = x##
    On the other you have ##f(x) = π/4 + ...\Sigma ... \cos((2n+1)x)\ + \ ... \sin ((2n+1)x)##
    Now make sure you have the right multipliers and coefficients
    and pick an x that lets the cosines disappear and the sines be (-1)n
     
  10. Aug 19, 2015 #9
    i got π/4 + ( ( -1)n-1 ) cos(nx) ) /π*n2+( (-1)nsin(nx) ) /n
    with the an=(-1)n-1) / (πn2) because during the an calculation i get an=( cos(nπ)-cos(n0) ) / (πn2) and cos1π=-1 and cos(0)=cos2π=1 so -1n=+1 for n=even and =-1 for n=odd
    and bn= - ( (-1)n ) / n from ( -nπcos(nπ) )/πn2

    I'm still unsure if I've made a mistake or not, as when i make x=π/4 i don't get 1 at n=0 like in the sequence but do get -0.37 for n=1. could that mean a mistake in the equation for the even term?
     
    Last edited: Aug 19, 2015
  11. Aug 20, 2015 #10

    BvU

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    Hi there,

    Yes, sorry, the (-1)n re-appears if you go back from summing 2n+1 to summing n .

    I've been piling error on top of error in this exercise (*) as well (nobody is perfect), but I convinced myself you are perfectly correct (I cheated by finding the series in a textbook example). Even made a plot for the first eight terms:

    Fourier_sawtooth.jpg

    Nice to see a triangle (no discontinuities) is easier to reproduce than a sawtooth !


    Now I'm not sure what you mean with "when i make x=π/4 i don't get 1 at n=0 like in the sequence"

    You make x a certain value and you add a good number of terms. At n = 0 you are supposed to get only the π/4 and the summation for the sines and cosines start at n = 1.

    Going back to your original exercise: to show π/4 = 1 - 1/3 + 1/5 you need a value for x that makes the cosine terms to go away. In fact the plot above makes the choice obvious ! So sorry for spoiling :wink: ...


    (*) E.g. the ##\ ... \sin ((2n+1)x)## in post #8 should have been ##\ ... \sin (nx)##
     
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