Needs["SpecialFunctions`"]

Übungsaufgabe

f=(1/((z^2)-3z))

Apart[f]

FPS[f,{z,3}]

FPS[f,{z,-1}]

Series[f,{z,1,10}]

FPS[f,{z,1}]

Series[f,{z,2I,10}]

FPS[f,{z,2I}]

Beispiel

Plot[Exp[-(1/(x^2))],{x,-0.1,0.1}]

Beispiel (4)

Series[((Sin [z])/z),{z,0,5}]

FPS[((Sin [z])/z),{z,0}]

Series[((Sin [z])/(z^2)),{z,0,5}]

FPS[((Sin [z])/(z^2)),{z,0}]

Series[(z/(Sin [z])),{z,0,5}]

(1/36)-(1/120)

Residuen

f=(1/((z^2)-3z))

Apart[f]

Series[f,{z,0,5}]

Residue[f,{z,0}]

Residue[f,{z,3}]

Residue[f,{z,5}]

Residue[(E^(1/z)),{z,0}]

Series[((Sin [z])/(z^4)),{z,0,2}]

Residue[((Sin [z])/(z^6)),{z,0}]

Series[((1+z)/(z-(z^3))),{z,0,5}]

Residue[((1+z)/(z-(z^3))),{z,0}]

Residue[((1+z)/(z-(z^3))),{z,1}]

Residue[((1+z)/(z-(z^3))),{z,-1}]

Apart[((1+z)/(z-(z^3)))]

Beispiel 4.13

f=((2(z^2)+z+1)/(((z^2)+1)(((z-2I))^2)))

Apart[f]

r1=Residue[f,{z,I}]

r2=Residue[f,{z,-I}]

r3=Residue[f,{z,2I}]

2π I(r1+r2+r3)//Simplify

2π I(r1+r2)//Simplify

Beispiel 4.14

f=((Sin[z])/(((((z^2)+1))^4)(((z+3I))^3)))

Apart[(f/(Sin[z]))]

Series[f,{z,I,2}]

r1=Residue[f,{z,I}]

r2=Residue[f,{z,-I}]

r3=Residue[f,{z,-3I}]

res=2π I(r1+r2+r3)//Simplify

res/.{Cosh[x_]->(((E^x)+(E^(-x)))/2),Sinh[x_]->(((E^x)-(E^(-x)))/2)}//Simplify

Beispiel 4.15

f=(1/((1+(z^2)+(z^4))))

Apart[f]

MapAll[ComplexExpand,Solve[(1/f)==0,z]]

r1=Residue[f,{z,-(1/2)+((I (Sqrt[3]))/2)}]

r2=Residue[f,{z,(1/2)+((I (Sqrt[3]))/2)}]

2π I(r1+r2)//Simplify

N [%]

Beispiel 4.16

f=((E^(I z))/((1+(z^2)+(z^4))))

Apart[f]

MapAll[ComplexExpand,Solve[(1/f)==0,z]]

r1=Residue[f,{z,-(1/2)+((I (Sqrt[3]))/2)}]

r2=Residue[f,{z,(1/2)+((I (Sqrt[3]))/2)}]

2π I(r1+r2)//ComplexExpand//Simplify

N [%]