Solve z=ln(x^2+y^2-1)

$$z = \ln ( x ^ { 2 } + y ^ { 2 } - 1 )$$

Solve for x (complex solution)

$$x=-\sqrt{1+e^{z}-y^{2}}$$

$$x=\sqrt{1+e^{z}-y^{2}}\text{, }Im(\ln(e^{z}))-Im(z)=0$$

Solve for y (complex solution)

$$y=-\sqrt{1+e^{z}-x^{2}}$$

$$y=\sqrt{1+e^{z}-x^{2}}\text{, }Im(\ln(e^{z}))-Im(z)=0$$

Solve for x

$$\left\{\begin{matrix}x=-\sqrt{1+e^{z}-y^{2}}\text{, }&\left(y\leq \sqrt{e^{z}+1}\text{ and }y>1\text{ and }y\geq e^{\frac{z}{2}}\text{ and }-\sqrt{1+e^{z}-y^{2}}\geq -1\right)\text{ or }\left(y\leq \sqrt{e^{z}+1}\text{ and }-\sqrt{1+e^{z}-y^{2}}<-1\text{ and }y>1\right)\text{ or }\left(y\geq -\sqrt{e^{z}+1}\text{ and }-\sqrt{1+e^{z}-y^{2}}<-1\text{ and }y<-1\right)\text{ or }\left(y\geq -\sqrt{e^{z}+1}\text{ and }y<-1\text{ and }y<-\sqrt{y^{2}-e^{z}}\text{ and }z\leq \ln(y^{2})\text{ and }-\sqrt{1+e^{z}-y^{2}}\geq -1\right)\\x=\sqrt{1+e^{z}-y^{2}}\text{, }&\left(y\leq \sqrt{e^{z}+1}\text{ and }y>1\text{ and }y\geq e^{\frac{z}{2}}\text{ and }\sqrt{1+e^{z}-y^{2}}\leq 1\right)\text{ or }\left(y\leq \sqrt{e^{z}+1}\text{ and }\sqrt{1+e^{z}-y^{2}}>1\text{ and }y>1\right)\text{ or }\left(y\geq -\sqrt{e^{z}+1}\text{ and }\sqrt{1+e^{z}-y^{2}}>1\text{ and }y<-1\right)\text{ or }\left(y\geq -\sqrt{e^{z}+1}\text{ and }y<-1\text{ and }y<-\sqrt{y^{2}-e^{z}}\text{ and }z\leq \ln(y^{2})\text{ and }\sqrt{1+e^{z}-y^{2}}\leq 1\right)\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=-\sqrt{1+e^{z}-x^{2}}\text{, }&\left(x\leq \sqrt{e^{z}+1}\text{ and }x>1\text{ and }x\geq e^{\frac{z}{2}}\text{ and }-\sqrt{1+e^{z}-x^{2}}\geq -1\right)\text{ or }\left(x\leq \sqrt{e^{z}+1}\text{ and }-\sqrt{1+e^{z}-x^{2}}<-1\text{ and }x>1\right)\text{ or }\left(x\geq -\sqrt{e^{z}+1}\text{ and }-\sqrt{1+e^{z}-x^{2}}<-1\text{ and }x<-1\right)\text{ or }\left(x\geq -\sqrt{e^{z}+1}\text{ and }x<-1\text{ and }x<-\sqrt{x^{2}-e^{z}}\text{ and }z\leq \ln(x^{2})\text{ and }-\sqrt{1+e^{z}-x^{2}}\geq -1\right)\\y=\sqrt{1+e^{z}-x^{2}}\text{, }&\left(x\leq \sqrt{e^{z}+1}\text{ and }x>1\text{ and }x\geq e^{\frac{z}{2}}\text{ and }\sqrt{1+e^{z}-x^{2}}\leq 1\right)\text{ or }\left(x\leq \sqrt{e^{z}+1}\text{ and }\sqrt{1+e^{z}-x^{2}}>1\text{ and }x>1\right)\text{ or }\left(x\geq -\sqrt{e^{z}+1}\text{ and }\sqrt{1+e^{z}-x^{2}}>1\text{ and }x<-1\right)\text{ or }\left(x\geq -\sqrt{e^{z}+1}\text{ and }x<-1\text{ and }x<-\sqrt{x^{2}-e^{z}}\text{ and }z\leq \ln(x^{2})\text{ and }\sqrt{1+e^{z}-x^{2}}\leq 1\right)\end{matrix}\right.$$