There are lots of tutorials about Fourier series online... google found these for me: http://www.facstaff.bucknell.edu/mastascu/eLessonsHTML/Freq/Freq4.html http://www.ecst.csuchico.edu/~amk/foo/grads/java/loyd/ http://www.falstad.com/fourier/ http://en.wikipedia.org/wiki/Fourier_series To make the calculations easier let's consider the case where the repeat-period L=2 Pi, then the formula for the Fourier coefficient a_m is: 1/Sqrt[2 Pi] Integrate[f[x] Exp[-I m x],{x,0,2 Pi}] Note that since the function is periodic, we are free to select a more centered interval: (-Pi,Pi): 1/Sqrt[2 Pi] Integrate[f[x] Exp[-I m x],{x,-Pi,Pi}] Find formulas for the Fourier coefficents for the functions: 1) f(x)=0 except in the range (-c,c) where it is 1, i.e., a symmetric square unit-high pulse that happens around x=0 Note that since this function is real and symmetric we expect a_m to be real, which means an expansion with just Cos (no Sin). 2) f(x)=0 except in the range (-c,c) where it equals Cos[Pi x/(2 c)] i.e., a symmetric unit-high cosine pulse that happens around x=0 and smothly goes to zero at x=+-c (as Cos[Pi/2]=0). Note that since this function is real and symmetric we expect a_m to be real, which means an expansion with just Cos (no Sin). For one of these cases find numerical values for the first four a_m that are non-zero in the case c=Pi/2 and convert the series to Cosine form and plot the resulting four-term sum in the range (-2 Pi,2 Pi).