(* source.nb *)

m=1

Clear[f];f[z_]:=m Log[z]

ComplexExpand[f[x+I y]]

phi[x_,y_]:=ComplexExpand[Re[f[x+I y]]]

psi[x_,y_]:=ComplexExpand[Im[f[x+I y]]]

g1=ContourPlot[phi[x,y]==Table[Log[r],{r,0.1,3.0,0.2}],{x,-2,2},{y,-2,2},ContourStyle->Directive[Red,Thin]]

g2=ContourPlot[psi[x,y]==Table[t,{t,-Pi,Pi,Pi/10}],{x,-2,2},{y,-2,2},ContourStyle->Directive[Blue,Thin]]

Show[g1,g2]

vv[x_,y_]:={Re[f'[x+I y]],-Im[f'[x+I y]]}

(*次はうまく行かない*)

g3=VectorPlot[vv[x,y],{x,-2,2},{y,-2,2},VectorScaling->Automatic]

(*原点の近くで大きいから、他が短くなってしまう*)

g3=VectorPlot[vv[x,y] Boole[x^2+y^2>0.2],{x,-2,2},{y,-2,2},VectorScaling->Automatic]

g4=Show[g1,g2,g3]

Export["source.png",g4]

Export["source.pdf",g4]