Funktionentheorie

u=((1-(x^2)-(y^2))/(1+(x^2)+(y^2)-2x))

D[u,x]//Together

D[u,{x,2}]//Together

D[u,{x,2}]+D[u,{y,2}]//Together

u=Exp[(x^2)-(y^2)]*Cos[2 x y]

D[u,{x,2}]//Together

D[u,{x,2}]+D[u,{y,2}]//Together

Umkehrfunktion der Joukowski-Funktion

Needs["Graphics`ComplexMap`"]

sol=Solve[w==(1/2)(z+(1/z)),z]

plot1=CartesianMap[z/.sol[[1]]/.w->#&,{-2,2},{-1,1}]

plot2=CartesianMap[z/.sol[[2]]/.w->#&,{-2,2},{-1,1}]

Show[plot1,plot2]

Kurvenintegrale

(∫_K^□)(((z-z0))^n)dz

int=(∫_0^(2π))((((z[t]-z0))^n)z'[t]/.{z->(z0+r (E^(I#))&)})\[DifferentialD]t

int=(∫_0^(2π))((((z[t]-z0))^(-1))z'[t]/.{z->(z0+r (E^(I#))&)})\[DifferentialD]t