Fourier and Laplace
Transforms

Dennis Silverman
Mathematical Physics 212A
Department of Physics and Astronomy
U.C. Irvine

<< "Calculus`FourierTransform`"

?FourierTransform

?InverseFourierTransform

FourierTransform[Exp[-(a^2) (t^2)], t, w]

PowerExpand[InverseFourierTransform[%, w, t]]

FourierTransform[1, t, w]

?NFourierTransform

<< "Calculus`LaplaceTransform`"

?LaplaceTransform

LaplaceTransform[1, t, s]

LaplaceTransform[DiracDelta[x - x0], x, s]

?UnitStep

LaplaceTransform[Sin[a x], x, s]

LaplaceTransform[Cos[a x], x, s]

LaplaceTransform[(x^n), x, s]

LaplaceTransform[Exp[-a x], x, s]

LaplaceTransform[f'[x],x,s]

LaplaceTransform[f''[x],x,s]

LaplaceTransform[(∫_0^x) f[t] \[DifferentialD]t, x, s]

Conv[x_] := (∫_0^x) f1[t] f2[x - t] \[DifferentialD]t

LaplaceTransform[Conv[x], x, s]

LaplaceTransform[f1[x] f2[x], x, s]

LaplaceTransform[x f[x], x, s]

LaplaceTransform[(x^2) f[x], x, s]

?InverseLaplaceTransform

InverseLaplaceTransform[(1/(((s^2) + 1) (s - 1))), s, t]

\(LaplaceTransform[∂\_{x, 2}u[x] == \(-k\^2\)\ u[x], x, s]\)

Null

ltu[s_]:=((s u[0]-u'[0])/((s^2) + (k^2)))

InverseLaplaceTransform[ltu[s], s, x]