Notatki z 4. ćwiczeń z MMF 14.03.2019

la1=Table[1+I a,{a,0,π,0.01}];

la1=Table[1+I a,{a,0,π,0.01}];
GraphicsRow[{ListPlot[{Re@#,Im@#}&/@(la1),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1],ListPlot[{Re@#,Im@#}&/@(Exp@la1),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1]}]

la2=Table[I π/2+a,{a,-10,10,0.01}];
GraphicsRow[{ListPlot[{Re@#,Im@#}&/@(la2),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1],ListPlot[{Re@#,Im@#}&/@(Exp@la2),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1]}]

la=Table[Cos[α]+I Sin[α],{α,0,2π,0.01}];
GraphicsRow[{ListPlot[{Re@#,Im@#}&/@(la),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1],ListPlot[{Re@#,Im@#}&/@(Exp@la),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1]}]

lb1=Table[π+I a,{a,0,1,0.01}];
GraphicsRow[{ListPlot[{Re@#,Im@#}&/@(lb1),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1],ListPlot[{Re@#,Im@#}&/@(Exp[I#]&@lb1),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1]}]

lc=Table[I a,{a,-10,10,0.01}];
GraphicsRow[{ListPlot[{Re@#,Im@#}&/@(lc),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1],ListPlot[{Re@#,Im@#}&/@(I Conjugate[#] &@lc),PlotTheme->"Scientific",PlotRange->{{-4,4},{-4,4}},AspectRatio->1]}]

Integrate[t Exp[-s t],t]

Integrate[Exp[-s t] t Sin[ω t],{t,0,∞}]

Integrate[(((1-(Sqrt[t])))^(1/3)),{t,0,1}]

Integrate[(((t-1))^(1/3))(((t+1))^(-1/2)),{t,-1,1}]
%//FullSimplify
N@%

(((-1))^(1/3))(2^(5/6))Gamma[4/3]((Gamma[1/2])/(Gamma[11/6]))//FullSimplify
%//N

Table[Gamma[-n+1/2],{n,10}]

Table[(((Sqrt[π])(2^n)(((-1))^n))/((2n-1)!!)),{n,10}]

Manipulate[Plot[Log@Binomial[n,x],{x,0,n},PlotTheme->"Scientific"],{{n,5},1,10,1}]

D[x Log[x/Exp[1]],x]//FullSimplify

Integrate[(1/(2+Sin[p])),{p,0,2π}]

Solve[(z^2)+4I z-1==0,z]