Cvičení 1: Funkce sinc(t)

Funkce sinc

sinc[t_/;t!=0.0]:=Sin[t]/t;
sinc[0.0]:=1.0;

Plot[sinc[t],{t,-10,10}]

Integrální sinus

Plot[SinIntegral[x],{x,0,20},PlotRange->{0,2},Ticks->{{0},{Pi/2}}]

Plot[SinIntegral[x],{x,-20,20},PlotRange->{-2,2},Ticks->{{0},{Pi/2}}]

Integrate[Sin[x]/x,{x,0,Infinity}]

sih[x_]:=Integrate[Sin[t]/t,{t,-Infinity,x}]

Plot[sih[x],{x,-20,20},PlotRange->{-0.5,4},Ticks->{{0},{Pi/2}}]

Mathematica má vestavěnou funkci pro integrální sinus SinIntegral[x]

?SinIntegral

Rozviňte funkci sinc(x) a Si(x) v Taylorovu řadu

Series[Sin[x]/x,{x,0,10}]

Series[SinIntegral[x],{x,0,10}]

Jednotková funkce

jed[t_/;t<0.0]:=0.0
jed[t_/;t>0.0]:=1.0
jed[0.0]:=0.5

Plot[jed[t],{t,-3,3},PlotRange->{-0.5,1.5},Ticks->{{1},{-1,1}},
PlotStyle->{Thickness[0.01]}]

Plot[jed[t-2],{t,-3,3},PlotRange->{-0.5,1.5},Ticks->{{1},{-1,1}},
PlotStyle->{Thickness[0.01]}]

Plot[(jed[t]-jed[t-2]),{t,-3,3},PlotRange->{-0.5,1.5},
     Ticks->{{1},{-1,1}},PlotStyle->{Thickness[0.01]}]

Plot[t*(jed[t]-jed[t-2]),{t,-3,3},PlotRange->{-0.5,2.5},
     Ticks->{{1},{-1,1}},PlotStyle->{Thickness[0.01]}]

Plot[2*(jed[t-2]),{t,-1,5},PlotRange->{-0.5,2.5},
     Ticks->{{1,2},{2,1}},PlotStyle->{Thickness[0.01]}]

Plot[t*(jed[t]-jed[t-2])+2*(jed[t-2]),{t,-1,5},PlotRange->{-0.5,2.5},
     Ticks->{{1,2},{2,1}},PlotStyle->{Thickness[0.01]}]