\Re(z)<0
z \mapsto \exp(z)-1
\Re(z)>0
z \mapsto \ln(z+1)
\alpha(z)
\alpha(z)+k
\Re(z)<0
k=\frac{-\pi i}{3}
\frac{\ln(z)}{3}
\frac{\ln(-z)}{3}
\Re(z)<0
z \mapsto \exp(z)-1
\Re(z)>0
z \mapsto \ln(z+1)
\alpha(z)
\alpha(z)+k
\Re(z)<0
k=\frac{-\pi i}{3}
\frac{\ln(z)}{3}
\frac{\ln(-z)}{3}
f(z)=\exp(z)-1
\alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\;
f(z)=\eta^z
f(z)=\exp(z)-1\;\;\;
\ln(\ln(\eta^{\eta^z}))
f(z)=\exp(z)-1
\alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\;
f(z)=\eta^z
f(z)=\exp(z)-1\;\;\;
\ln(\ln(\eta^{\eta^z}))
f(z)=\exp(z)-1
\alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\;
f(z)=\eta^z
f(z)=\exp(z)-1\;\;\;
\ln(\ln(\eta^{\eta^z}))