Math Tables: Fourier Series


Useful Papers:
The fourier series of the function f(x)

\[ \frac{a(0)}{2} + \sum<sub>k=1</sub>^\infty (a(k) \cos kx + b(k) \sin kx) \]

a(k) = 1/PI (integral)(-PI to PI) f(x) cos kx dx
b(k) = 1/PI (integral)(-PI to PI) f(x) sin kx dx
Remainder of fourier series. Sn(x) = sum of first n+1 terms at x.

remainder(n) = f(x) - Sn(x) = 1/PI (integral)(-PI to PI) f(x+t) Dn(t) dt

Sn(x) = 1/PI (integral)(-PI to PI) f(x+t) Dn(t) dt

Dn(x) = Dirichlet kernel = 1/2 + cos x + cos 2x + .. + cos nx = [ sin(n + 1/2)x ] / [ 2sin(x/2) ]
Riemann's Theorem. If f(x) is continuous except for a finite # of finite jumps in every finite interval then:

lim(k->inf) (integral)(a-b) f(t) cos kt dt = lim(k->inf)(integral)(a-b)f(t) sin kt dt = 0

The fourier series of the function f(x) in an arbitrary interval.

A(0) / 2 + (sum)(k=1..inf) [ A(k) cos (k(PI)x / m) + B(k) (sin k(PI)x / m) ]

a(k) = 1/m (integral)(-m-&gtm)f(x) cos (k(PI)x / m) dx
b(k) = 1/m (integral)(-m-&gtm)f(x) sin (k(PI)x / m) dx
Parseval's Theorem. If f(x) is continuous; f(-PI) = f(PI) then

1/PI (integral)(-PI to PI) f2(x) dx = a(0)2 / 2 + (sum)(k=1..inf) (a(k)2 + b(k)2)

Fourier Integral of the function f(x)

f(x) = (integral)(0-inf) ( a(y) cos yx + b(y) sin yx ) dy

a(y) = 1/PI (integral)(-inf-&gtinf) f(t) cos ty dt
b(y) = 1/PI (integral)(-inf-&gtinf) f(t) sin ty dt
f(x) = 1/PI (integral)(0-inf) dy (integral)(-inf-&thinf)f(t) cos (y(x-t)) dt

Special Cases of Fourier Integral

if f(x) = f(-x) then

f(x) = 2/PI (integral)(0-inf) cos xy dy (integral)(0-inf)f(t) cos yt dt
if f(-x) = -f(x) then
f(x) = 2/PI (integral)(0-inf) sin xy dy (integral)(0-inf)sin yt dt
Fourier Transforms

Fourier Cosine Transform

g(x) = sqrt(2/PI)(integral)(0-inf)f(t) cos xt dt

Fourier Sine Transform

g(x) = sqrt(2/PI)(integral)(0-inf)f(t) sin xt dt

Identities of the Transforms

If f(-x) = f(x) then

Fourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x)
If f(-x) = -f(x) then
Fourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)