arctanh(z)

[Tleko]

arctanh(z) = .5*ln((1+z)/(1-z)), z=x+i*y

Real = ln(((1-x^2-y^2)^2)/((1-x)^2+y^2))

Imag.= .5*atan(2*y/(1-x^2-y^2))

(using MATLAB):

Realebene

[x,y]=meshdom(0:.05:1,0:.05:1);
colormap([1 1 1]);
R=log(((1-x.^2-y.^2).^2+4*y.^2)./((1-x).^2+y.^2));
I=.5*atan(2*y./(1-x.^2-y.^2));
z=sqrt(R.^2+I.^2).*cos(atan(I./R));
surf(x,y,z)

Imaginaerebene

[x,y]=meshdom(0:.05:1,0:.05:1);
colormap([1 1 1]);
R=log(((1-x.^2-y.^2).^2+4*y.^2)./((1-x).^2+y.^2));
I=.5*atan(2*y./(1-x.^2-y.^2));
z=sqrt(R.^2+I.^2).*sin(atan(I./R));
surf(x,y,z)

tl...@aol.com

[Tleko]

Im Artikel <199809081228...@ladder03.news.aol.com>
tl...@aol.com (Tleko)

Da sind einige Fehler in meinem Artikel unterlaufen. Ich sollte schreiben:

arctanh(z) = .5*ln((1+z)/(1-z)), z=x+i*y

Real = .5*ln(sqrt(((1-x^2-y^2)^2+(2*y)^2)/(((1-x)^2+y^2))^2))

Imag.= .5*atan(2*y/(1-x^2-y^2))

(using MATLAB):

Realebene

[x,y]=meshdom(0:.05:1,0:.05:1);
colormap([1 1 1]);

R=.5*log(sqrt(((1-x.^2-y.^2).^2+(2*y).^2./(((1-x).^2+y.^2)).^2));
I=.5*atan((2*y)./(1-x.^2-y.^2));
z=sqrt(R.^2+I.^2).*cos(atan(I./R));
surf(x,y,z)

Imaginaerebene

[x,y]=meshdom(0:.05:1,0:.05:1);
colormap([1 1 1]);

R=.5*log(sqrt(((1-x.^2-y.^2).^2+(2*y).^2./(((1-x).^2+y.^2)).^2));
I=.5*atan((2*y)./(1-x.^2-y.^2));
z=sqrt(R.^2+I.^2).*sin(atan(I./R));
surf(x,y,z)

tl...@aol.com