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Using Parserval's identity in a Fourier series

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$begingroup$

I have the function $$f(x)=begincases1, space |x|leq a \ 0, space a<|x|leq 1/2endcases$$
I have calculated the Fourier series of this even function: $$Sf(x)=2a+2sum_n=0^infty fracsin(2pi na)pi ncos(2pi nx)$$
Now I need to calculate $$sum_n=0^inftyfracsin^2(2pi na)n^2$$
How do I use Parserval's Identity to calculate this last sum? I have tried $$2a=int_-a^a1^2=2a^2+sum_n=0^inftyfracsin^2(2pi na)pi^2 n^2$$ And therefore $$(2a-2a^2)pi^2=sum_n=0^inftyfracsin^2(2pi na)n^2$$
Is this correct? If not, what is the correct way to do it?

fourier-series

share|cite|improve this question

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Those sums should probably run from $1$ to $infty$; you don't want a literal $frac00$ term in there, and the zero term $2a$ has a different formula anyway.
  $endgroup$
  – jmerry
  Jan 30 at 0:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The leading constant should be part of the first fourier coefficient to have the form $Sf = sum_n c_n cos(2pi nx)$ for which we can apply Parserval's identity.
  $endgroup$
  – Nicholas Parris
  Jan 30 at 0:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @NicholasParris How would you plug in the leading constant into the sum? I'm also interested in this problem
  $endgroup$
  – John Keeper
  Feb 8 at 1:55
  

  
  
  
  

add a comment | 

0

$begingroup$

I have the function $$f(x)=begincases1, space |x|leq a \ 0, space a<|x|leq 1/2endcases$$
I have calculated the Fourier series of this even function: $$Sf(x)=2a+2sum_n=0^infty fracsin(2pi na)pi ncos(2pi nx)$$
Now I need to calculate $$sum_n=0^inftyfracsin^2(2pi na)n^2$$
How do I use Parserval's Identity to calculate this last sum? I have tried $$2a=int_-a^a1^2=2a^2+sum_n=0^inftyfracsin^2(2pi na)pi^2 n^2$$ And therefore $$(2a-2a^2)pi^2=sum_n=0^inftyfracsin^2(2pi na)n^2$$
Is this correct? If not, what is the correct way to do it?

fourier-series

share|cite|improve this question

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Those sums should probably run from $1$ to $infty$; you don't want a literal $frac00$ term in there, and the zero term $2a$ has a different formula anyway.
  $endgroup$
  – jmerry
  Jan 30 at 0:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The leading constant should be part of the first fourier coefficient to have the form $Sf = sum_n c_n cos(2pi nx)$ for which we can apply Parserval's identity.
  $endgroup$
  – Nicholas Parris
  Jan 30 at 0:25
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @NicholasParris How would you plug in the leading constant into the sum? I'm also interested in this problem
  $endgroup$
  – John Keeper
  Feb 8 at 1:55
  

  
  
  
  

add a comment | 

$begingroup$

I have the function $$f(x)=begincases1, space |x|leq a \ 0, space a<|x|leq 1/2endcases$$
I have calculated the Fourier series of this even function: $$Sf(x)=2a+2sum_n=0^infty fracsin(2pi na)pi ncos(2pi nx)$$
Now I need to calculate $$sum_n=0^inftyfracsin^2(2pi na)n^2$$
How do I use Parserval's Identity to calculate this last sum? I have tried $$2a=int_-a^a1^2=2a^2+sum_n=0^inftyfracsin^2(2pi na)pi^2 n^2$$ And therefore $$(2a-2a^2)pi^2=sum_n=0^inftyfracsin^2(2pi na)n^2$$
Is this correct? If not, what is the correct way to do it?

fourier-series

share|cite|improve this question

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

$endgroup$

share|cite|improve this question

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

share|cite|improve this question

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

edited Mar 21 at 12:42

Rócherz

3,0013822

edited Mar 21 at 12:42

Rócherz

3,0013822

asked Jan 30 at 0:11

codingnightcodingnight

927

asked Jan 30 at 0:11

codingnightcodingnight

927