Lösungen Laplace-Rückransformationen

?*Laplace*

?Laplace*

?InverseLaplace*

?Limit

?Direction

Remove["Global`*"]

1

a

f[t_]:= t Sin[t]

transf= LaplaceTransform[f[t],t,s]

Limit[f[t],t->Infinity]

Limit[s transf,s->0,Direction->-1]

!! Ungleiches Resultat - Wieso? !!

Limit[f[t],t->0,Direction->-1]

Limit[s transf,s->Infinity]

b

f[t_]:= 1/t Sin[t]

transf= LaplaceTransform[f[t],t,s]

Limit[f[t],t->Infinity]

Limit[s transf,s->0,Direction->-1]

Limit[f[t],t->0,Direction->-1]

Limit[s transf,s->Infinity]

c

f[t_]:= Integrate[1/τ Sin[τ],{τ,0,t}]

transf= LaplaceTransform[Evaluate[f[t]],t,s]

Limit[f[t],t->Infinity]

Limit[s transf,s->0,Direction->-1]

Limit[f[t],t->0,Direction->-1]

Limit[s transf,s->Infinity]

2

a

InverseLaplaceTransform[(6 s + 2)/(2 s^2+4 s+8),s,t]

b

InverseLaplaceTransform[2/(2 s^2+4 s+8),s,t]

c

InverseLaplaceTransform[(6 s - 2)/(2 s^2-4 s-8),s,t]

d

InverseLaplaceTransform[(6 s - 2)/(2 s^2+4 s+8)^2,s,t]

3

a

InverseLaplaceTransform[(3 s + 4)/(s^2+2 s-3),s,t]

b

InverseLaplaceTransform[2/(s+2)-2s/(s^2+16)+3/(s^2+15),s,t]

c

InverseLaplaceTransform[1/(s^2+4 s+3),s,t]

d

InverseLaplaceTransform[e^(- Pi/4 s) s/(s^2-4),s,t]

(Handrechnung, Verschiebung der Originalfunktion um, f(t)=0 für t< π/4 , sonst um π/4 verschobene Rücktransformierte von s/(s^2-4)). Diese ist von der Verschiebung:

InverseLaplaceTransform[s/(s^2-4),s,t]

Verschiebung:

(1/2) (E^(-2 t)) (1 + (E^(4 t))) /. t -> (t - Pi/4)

e

InverseLaplaceTransform[(2s-6)/(s^2+4s-20),s,t]

f

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

g

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