# Thread: Integrating fourier series

1. ## Integrating fourier series

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)
$\displaystyle \pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)$

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

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

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

but this is where it gets hard for me,

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

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.

2. 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)
$\displaystyle \pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)$

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

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

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

but this is where it gets hard for me,

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

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?

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

4. 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.

5. 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.