Fourier Series & Transform
Inverse Fourier Transform of a Fourier Transform F
f(x) = (1/√2π) × the integral over an infinite domain of the fourier transform F(ε) × exp(iεx) dε
Fourier Sine and Cosine Series Representation
f(x) = ao/2 + ∑↓n=1↑∞ [an cos(nπx/a) + bn sin(nπx/a)], .where the coefficients are defined equivalently to the separate definitions
Fourier Cosine Series Representation for an Even Function φ(x)
in the combination bn = 0 and an are defined as : an = 2/a ∫↓0↑a φ(x) cos(nπx/a) dx
Cauchy's Residue Theorem
the value of an integral around a closed contour is 2πi times the sum of the residues at the pole inside the contour. COMPLEX FUNCTIONS THEOREM
Time when Fourier Transforms cannot be used
where f(x) = constant, exp(x), sin(x) Forier Transforms cannot be used as the integral does not exist. When time is defined over a (semi) infinite domain Fourier Transforms cannot be used.
Fourier Sine Series Representation of a suitable function with Fourier Sine Coefficients (Odd function) ( an = 0 )
∑↓n=1↑∞ bn sin (nπx/a) =φ(x) for 0<x<a, with an given by : bn = 2/a ∫↓o↑a φ(x) sin (nπx/a) dx
Fourier Transform of a function f
F(ε) = (1/√2π) × the integral over an infinite domain of the function f(x) × exp(-iεx) dx
Linearity of Fourier Transforms
F[ af + bg ] = aF[f] + bF[g]
Convolution Property
F[f*g] = F[f]F[g] where (f*g)(x) = 1/√2π × integral over an infinite domain of f(x-ε)g(ε) dε
Partial Derivatives acting on Fourier Transforms
F[ux] = iεF[u], F[uxx] = -ε²F[u] F[ut] = d/dt F[u] F[utt] = d²/dt² F[u]
Addition of a constant to a function (shift property)
For g(x) = f(x+a), The fourier transform of g is equal to the fourier transform of f times exp(ika)
Properties of Real/Imaginary Functions
If f is real and even the the fourier transform of f is real. If f is real and odd the the fourier transform of f is imaginary.
Dirichlet's Theorem for a function to have a Fourier Series Representation
If f(x) is a bounded periodic function that contains a finite number of maxima, minima and points of discontinuity in each period then the Fourier series of f(x) converges to f(x) at each x where f(x) is continuous and to the average of surrounding points where f(x) is discontinuous.
