Integrating fourier series

**olski1** · May 10th 2011, 02:52 AM

ok so i had to find the fourier series of cos(ax) with period 2 pi where a is not an integer.

So i came to the correct answer iam pretty sure, then they asked me to sub in x =pi into the series to get
i)
$\pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)$

$\pi cot(\pi a) -\frac{1}{a } = 2a(\frac{1}{{a}^{ 2} -{1}^{2 } }+ \frac{1}{{a}^{ 2} -{2}^{2}} ........)$

ii)now i have to integrate both sides form a =0 to a=theta to obtain

$\frac{\sin \theta \pi }{\theta \pi }=(1-\frac{{\theta }^{2 } }{{1}^{ 2}.....) }$

but this is where it gets hard for me,

integrating the left hand side of i) i get
$\frac{ln(sin(\pi a)}{\pi } - \frac{lnsin(a\pi )}{\ pi }$

but when calculating from a=0 isnt it undefined?

I dont even know where to start on the right hand side of the equation so any help with that would be greatly appreciated.

**mr fantastic** · May 10th 2011, 04:40 AM

Originally Posted by olski1

ok so i had to find the fourier series of cos(ax) with period 2 pi where a is not an integer.

So i came to the correct answer iam pretty sure, then they asked me to sub in x =pi into the series to get
i)
$\pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)$

$\pi cot(\pi a) -\frac{1}{a } = 2a(\frac{1}{{a}^{ 2} -{1}^{2 } }+ \frac{1}{{a}^{ 2} -{2}^{2}} ........)$

ii)now i have to integrate both sides form a =0 to a=theta to obtain

$\frac{\sin \theta \pi }{\theta \pi }=(1-\frac{{\theta }^{2 } }{{1}^{ 2}.....) }$

but this is where it gets hard for me,

integrating the left hand side of i) i get
$\frac{ln(sin(\pi a)}{\pi } - \frac{lnsin(a\pi )}{\ pi }$

but when calculating from a=0 isnt it undefined?

I dont even know where to start on the right hand side of the equation so any help with that would be greatly appreciated.

Is this part of an assignment that counts towards your final grade?

**olski1** · May 10th 2011, 04:41 AM

okay so figured out left hand side of the equation. But how do i go about integrating the right hand side to get
$\int 2a(\frac{1}{{a}^{ 2} -{1}^{2 } }+ \frac{1}{{a}^{ 2} -{2}^{2}} ........)= (1-\frac{\theta ^2}{ 1^2}).....?$

**olski1** · May 10th 2011, 05:57 AM

Originally Posted by mr fantastic

Is this part of an assignment that counts towards your final grade?

Yes it is, did not want answers just guidance. But does not matter any more. It took me a while, but I solved it.

**mr fantastic** · May 10th 2011, 03:39 PM

Originally Posted by olski1

Yes it is, did not want answers just guidance. But does not matter any more. It took me a while, but I solved it.

Thankyou for this honest reply.

Thread: Integrating fourier series

LinkBack

Thread Tools

Display

Integrating fourier series

Similar Math Help Forum Discussions

Integrating a fourier series

[SOLVED] Fourier series to calculate an infinite series

Integrating Fourier Series

Complex Fourier Series & Full Fourier Series

from fourier transform to fourier series