Math2.org Math Tables: Fourier Series
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The fourier series of the function f(x)
![\[ \frac{a(0)}{2} + \sum<sub>k=1</sub>^\infty (a(k) \cos kx + b(k) \sin kx) \]](/math/tex/7c2a4392ed04bb74303f5b8b0bc1447a.png "\[ \frac{a(0)}{2} + \sum<sub>k=1</sub>^\infty (a(k) \cos kx + b(k) \sin kx) \]")
a(k) = 1/PIf(x) cos kx dx
b(k) = 1/PI
Remainder of fourier series. Sn(x) = sum of first n+1 terms at x.
remainder(n) = f(x) - Sn(x) = 1/PI 
f(x+t) Dn(t) dt
Sn(x) = 1/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->
) 
f(t) cos kt dt = lim(k->)f(t) sin kt dt = 0
The fourier series of the function f(x) in an arbitrary interval.
A(0) / 2 +
1/PI 
(k=1..) [ A(k) cos (k(PI)x / m) + B(k) (sin k(PI)x / m) ]
a(k) = 1/m
f(x) cos (k(PI)x / m) dxb(k) = 1/m
f(x) sin (k(PI)x / m) dx Parseval's Theorem. If f(x) is continuous; f(-PI) = f(PI) then f2(x) dx = a(0)2 / 2 + (k=1..) (a(k)2 + b(k)2)
Fourier Integral of the function f(x)
f(x) = 
( a(y) cos yx + b(y) sin yx ) dy
a(y) = 1/PI
f(t) cos ty dtb(y) = 1/PI
f(x) = 1/PI f(t) sin ty dt dy f(t) cos (y(x-t)) dt
Special Cases of Fourier Integral
if f(x) = f(-x) then
f(x) = 2/PI
if f(-x) = -f(x) then
cos xy dy f(t) cos yt dtf(x) = 2/PI
sin xy dy sin yt dt Fourier Transforms
Fourier Cosine Transform
g(x) = 
(2/PI)f(t) cos xt dt
Fourier Sine Transform
g(x) = (2/PI)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)
