(1)Laplaceovo desatero Fourierovo desatero L[f(ct

Laplaceovo desatero Fourierovo desatero L[f(ct)] = ¹_cF(^p_c) F[f(c⃗x)] = _|c¹_|rF(^ξ⃗_c) L[(−t)ⁿf(t)] =^d_dp^nFn F[(i⃗x)^αf(⃗x)] =D^αF[f(⃗x)]

L[ ˙f(t)] =pL[f(t)]−f(0+) F[D^α( f(⃗x))

] = (−i⃗ξ)^αF[f(⃗x)]

L[Θ(t)∫t

0 f(τ)dτ] = ^F(p)_p F[1] = (2π)^rδ(⃗ξ) L[^f(t)_t ] =∫_∞

p F(q)dq FF[f(x)] = (2π)^rf(−x) e^apL[f(t)] =L[f(t+a)] e^i⃗µ⃗ξF[f(⃗x)] =F[f(⃗x−⃗µ)]

L[e^atf(t)] =F(p−a) F[e^i⃗µ⃗xf(⃗x)] =F(⃗ξ+⃗µ)

∫_∞

0 f(τ)dτ = lim_p→0+F(p) lim_|ξ|→∞F(ξ) = 0 L[f(t) ⋆ g(t)] =F(p)·G(p) F[f(⃗x)⋆ g(⃗x)] =F[f(⃗x)]·F[g(⃗x)]

∫_∞

0 f(t)G(t)dt=∫_∞

0 F(t)g(t)dt ∫_∞

−∞f(x)G(x)dx=∫_∞

−∞F(x)g(x)dx

Laplace·v vzor Laplace·v obraz δ(t−τ) e^−pτ

Θ(t) ¹_p

Θ(t)tⁿ (n∈N0) _pn+1^n!

Θ(t)t^α (α >−1) ^Γ(α+1)_pα+1

Θ(t)e^µt _p−¹_µ

Θ(t) sin(βt) _p2+β^{β 2}

Θ(t) cos(βt) _p2+β^{p 2}

Θ(t) (sin(t)−tcos(t)) _(1+p²2)²

Θ(t)e^µtcos(ωt) _(p−^p_µ)⁻2^µ+ω²

Θ(t)e^µtsin(ωt) _(p−µ)^ω_2+ω2 Θ(t) sinh(ωt) _p2−^ω_ω2 Θ(t) cosh(ωt) _p2−^pω²

Fourie·v vzor Fourie·v obraz Obor e^−a∥x∥2 (^π_a)^r/2e^−∥

⃗ξ∥2

4a E^r

Θ(x)e^ax, (a̸= 0) _a+iξ⁻¹ R

δ(⃗x−⃗µ) e^i⃗ξ⃗µ E^r

Θ(x) πδ(ξ) + iP¹_ξ R

Θ(−x) πδ(ξ) −iP¹_ξ R

sgn(x) 2iP¹_ξ R

1 (2π)^rδ(⃗ξ) E^r

P_x¹ iπsgn(ξ) R

P_x¹2 −π|ξ| R

e^ix2 √

πe^{−4i(ξ2−π)} R

Θ(R− |x|) 2^sin(Rξ)_ξ R

Θ(R−∥⃗x∥)

√R²−∥⃗x∥² 2π^{sin(R∥⃗ξ∥)}

∥⃗ξ∥ E² δSR(⃗x) 4πR^{sin(R∥⃗ξ∥)}

∥⃗ξ∥ E³

⃗

x^α (i)^|α|(2π)^rδ^(α)(ξ)⃗ E^r

e^icx 2πδ(ξ+c) R

cos(cx) π(

δ(ξ−c) +δ(ξ+c)) R

sin(cx) iπ(

δ(ξ−c)−δ(ξ+c)) R