fantastic fourier series
Created | Updated Mar 28, 2002
Warning - semi-unpalatable mathematics here.
Little header, integrals are with respect to t and are represented with Heaviside's Big D notation.
Fourier series are really nifty. Take any function f(x) and it can be represented in an infinite series of sines and cosines. How do we do it? We generate two series, one the coefficients of sines, the other, the coefficients of cosines. Call the former a set of coefficients S, the latter, the set C. Call the kth element of S S_k, the kth element of C C_k.
Fourier series work in a specific interval [-l,l], and sometimes in [-inf,inf]
Then:
C_0/2+C_1*cos((pi*x)/l)+S_1*sin((pi*x)/l)+C_2*cos((2*pi*x)/l)+S_2*sin((pi*x)/l)+C_3*cos((3*pi*x)/l)+...
The above can probably better be represented with sigma notation but hey, I'm no magician.
Anyhow, how do we create these two sets? We use:
C_k=1/l*D^(-1)(f(t)*cos((k*pi*t)/l))
S_k=1/l*D^(-1)(f(t)*sin((k*pi*t)/l))
Great, I hear you say, but the only thing is: for complicated functions these integrals get *messy*. So if you want to tinker with some fourier series then start easy! And have fun. Yes. Math is fun!
Little header, integrals are with respect to t and are represented with Heaviside's Big D notation.
Fourier series are really nifty. Take any function f(x) and it can be represented in an infinite series of sines and cosines. How do we do it? We generate two series, one the coefficients of sines, the other, the coefficients of cosines. Call the former a set of coefficients S, the latter, the set C. Call the kth element of S S_k, the kth element of C C_k.
Fourier series work in a specific interval [-l,l], and sometimes in [-inf,inf]
Then:
C_0/2+C_1*cos((pi*x)/l)+S_1*sin((pi*x)/l)+C_2*cos((2*pi*x)/l)+S_2*sin((pi*x)/l)+C_3*cos((3*pi*x)/l)+...
The above can probably better be represented with sigma notation but hey, I'm no magician.
Anyhow, how do we create these two sets? We use:
C_k=1/l*D^(-1)(f(t)*cos((k*pi*t)/l))
S_k=1/l*D^(-1)(f(t)*sin((k*pi*t)/l))
Great, I hear you say, but the only thing is: for complicated functions these integrals get *messy*. So if you want to tinker with some fourier series then start easy! And have fun. Yes. Math is fun!
