In the link below to my worksheet I have attempted to construct the Fourier series for the function 1-x.  I should mention that the I often interchnge the variables x & t (probably a bad habit).  It appears that I am getting something close, but my term for n=0 appears to be undefined.  Given the plot I generated the term for n=0 should be approximately 1/4 to get the base of the triangle to intersect the t or x axis.

So I thought if I can get another set of eyes on my work maybe someone can point out what I have incorrect.

Appreciate any assistance

asymmetric_triangle_fourier_series.mw

I must be missing something in my Fourier integral.  My understanding is that the sinc function is the transform of a square wave.  In the link below I am getting something slightly different.  I have the parameter tau to define relative to the period, T, to vary the width aspect ratio of the wave,  If tau=T I do get the sinc function.

What am I missing or is what I have correct?

Sq_wave_Fourier_transform.mw

The link below is to a worksheet giving the output of the sinc(0).  There is some contradictory results.  I know sinc(x)=sin(x)/x.  If x is very small then sinc(x)-->1.0 for either +-x.  However, for x = 0 is the function actually undefined or = 0 or = 1.0?  Check the link.  If you have an explanation I would like to be enlightened.

sinc(0).mw

Not sure what is going on, but I confirmed the result is incorrect via MATLAB.  Can anyone tell me what MAPLE is doing?  Evaluation of the 1st expression is correct, but the 2nd expression = 0 is incorrect.  The correct answer should be 1.862645149230957e-08.

discrepancy.mw

In the link below is my code which generates numerical data of a Fourier series of an impulse function (not to be confused with the DIRAC DELTA IMPULSE).  The issue is as k increases the harmonic frequencies become faster than the time step which causes resolution issues at higher k values.  Is there a way to iteratively change the time step for each iteration of k value?  However, the range for t would have to correspondingly decrease or the file size would balloon by the corresponding change in k value.

So lets say my time step for k = 1 is 0.001 and my time range is 0..1, now for k = 1000 my time step would be 0.000001, but my time range would shrink from 0..1 to 0..0.001.  I hope this is not too confusing.  Maybe experimenting with my code will aid in understanding my request.

Thanks for any assistance

MATRIX_loop.mw