I am following this post since I am going the same problem as SM.



Following the general equation as quoted from various reputable sources, such as John Bird, although the cos(x)/2 term perfect sense, I cannot understand it's origin. I have proved it graphically to be the missing term, but mathematically I cannot.

I did read through the limits part a number of times but it was above me. Please follow my reasoning and guide myself and SM through this.

Thanks for your help so far
Mike

ps. the graph x axis is plotted as a function of pi, i.e. 1 is actually pi, 0.5 is pi/2

 

Hi there,

Well, in your paper for 'an' you have what looks correct:
an=2*cos(n*pi/2)/(pi*(1-n^2))

and that has numerator:
N=2*cos(n*pi/2)

and denominator:
D=pi*(1-n^2)

and if we replace 'n' in the numerator with an odd number from 3 to infinity, or say 3 to 33, we always get zero,
and if we replace 'n' in this denominator with an odd number from 3 to infinity, or say from 3 to 33, we always get a real number that is not zero. So we end up with zero divided by some real number N:
an=0/N

which always leads to 'an' being equal to zero:
an=0

However, when we replace 'n' in the numerator with 1 we still get zero in the numerator, but when we replace 'n' in the denominator with 1 we also get zero, so we end up with this:
an=0/0

and since this has zero in the denominator that means we can not calculate this directly but must use the concept of limits to get the right result. We know we can not replace 'n' in the denominator with 1 but luckily there is an easy way to handle this using the derivative of the numerator and denominator.

First we take the derivative of the numerator with respect to 'n' and get:
dNum=-pi*sin(pi*n/2)

and then take the derivative of the denominator with respect to 'n' and get:
dDenom=-2*pi*n

now we reform the fraction:
dNum/dDenom=(-pi*sin(pi*n/2))/(-2*pi*n)

and now we take the limit of this as 'n' goes to 1. Now that we have just -2*pi*n in the denominator, we get a simpler result when we replace 'n' with 1, and we get 1 in the numerator and 2 in the denominator, so the result is:
1/2

and so the reconstruction term ends up being cos(x)/2.

So the short answer is that 'an' ends up being zero for odd numbers that are 3 or above, but ends up being non zero for n=1, and that's the only odd number where it is non zero.

In reality we have to check every 'n' to make sure they are really zero or not. We often take short cuts based on one or two calculations but when we do this we have to be very careful that we did not miss something.