# On Hyperbolic Complex Numbers
## 2 The Basic Hyperbolic Complex Algebraic Structure
### Hyperbolically Modified Cauchy-Riemann Differential Equations  
Let \(z_{k}\in\mathfrak{X},k=1,2\). According to Definition 4, we call \(z_{1}\) hyperbolic multiplicative reachable from \(z_{2}\) if there exists \(z_{3}\in\mathfrak{X}\) such that \(z_{1}=z_{2}\odot_{h}z_{3}\). If this is so, then we call \(z_{3}\) the hyperbolic ratio of \(z_{1}\) and \(z_{2}\) and write this as  
\[z_{3}=z_{1}\oslash_{h}z_{2}.\]  
One can check that  
\[z_{3}=\frac{1}{x_{2}^{2}-y_{2}^{2}}\left(\begin{array}{c}x_{1}x_{2}-y_{1}y_{ 2}\\ x_{2}y_{1}-x_{1}y_{2}\end{array}\right) \tag{13}\]  
and in particular  
\[z_{1}\oslash_{h}\left(\begin{array}{c}1\\ 0\end{array}\right)=z_{1}.\]  
If we apply (13) also for \(z_{1}=z\in\mathfrak{X}\) and \(z_{2}=(0,1)^{T}\in\mathfrak{Y}\), then  
\[z\oslash_{h}\left(\begin{array}{c}0\\ 1\end{array}\right)=-\left(\begin{array}{c}y\\ x\end{array}\right).\]  
While \(z\to z\oslash_{h}\left(\begin{array}{c}1\\ 0\end{array}\right)\) means the identity mapping, the mapping \(z\to z\oslash_{h}\left(\begin{array}{c}0\\ 1\end{array}\right)\) means reflection of the point \(z=\left(\begin{array}{c}x\\ y\end{array}\right)\) at the line \(y=-x\).  
Let \(D\) be an open subset of \(\mathfrak{X}\). A function \(f:D\to\mathfrak{X}\) is called hyperbolically differentiable in \(z_{0}\in D\) if the ratio \((f(z)-f(z_{0}))\oslash_{h}(z-z_{0})\) converges to a limit point \(f^{\prime}(z_{0})\), say, as \(z\to z_{0}\), that is  
\[f^{\prime}(z_{0})=\lim_{z\to z_{0}}(f(z)-f(z_{0}))\oslash_{h}(z-z_{0}).\]  
The function \(f\) will be called hyperbolically holomorphic in \(D\) if it is differentiable in every \(z_{0}\) from \(D\). If such function is given by  
\[f(\left(\begin{array}{c}x\\ y\end{array}\right))=\left(\begin{array}{c}u(x,y)\\ v(x,y)\end{array}\right),\left(\begin{array}{c}x\\ y\end{array}\right)\in D\]  
then it follows that hyperbolic modifications of the well known Cauchy-Riemann differential equations hold. To become more specific, we consider points \(z_{l}=z_{0}+z_{l,h},l=1,2\) with  
\[z_{1,h}=\left(\begin{array}{c}h\\ 0\end{array}\right)\text{ and }z_{2,h}=\left(\begin{array}{c}0\\ h\end{array}\right).\]  
Letting now the values of \(h\) tending to zero, and, respectively, denoting the partial derivatives with respect to \(x\) or \(y\) of a function \(w(x,y)\) by \(w_{x}\) and \(w_{y}\), we get  
\[(f(z_{1})-f(z_{0}))\oslash_{h}(z_{1}-z_{0}) =\left(\begin{array}{c}u(x_{0}+h,y_{0})-u(x_{0},y_{0})\\ v(x_{0}+h,y_{0})-v(x_{0},y_{0})\end{array}\right)\oslash_{h}z_{1,h}\] \[=\left(\begin{array}{c}u_{x}(x_{0}+\delta h,y_{0})\\ v_{x}(x_{0}+\delta h,y_{0})\end{array}\right)\oslash_{h}\left(\begin{array}[] {c}1\\ 0\end{array}\right)\to\left(\begin{array}{c}u_{x}(x_{0},y_{0})\\ v_{x}(x_{0},y_{0})\end{array}\right),h\to 0\]  
where \(\delta\) is a certain number from the interval \((0,1)\). Similarly,  
\[(f(z_{2})-f(z_{0}))\oslash_{h}(z_{2}-z_{0}) =\left(\begin{array}{c}u(x_{0}+h,y_{0})-u(x_{0},y_{0})\\ v(x_{0}+h,y_{0})-v(x_{0},y_{0})\end{array}\right)\oslash_{h}z_{2,h}\] \[=\left(\begin{array}{c}u_{y}(x_{0}+\delta h,y_{0})\\ v_{y}(x_{0}+\delta h,y_{0})\end{array}\right)\oslash_{h}(0,1)^{T}\to\left( \begin{array}{c}-v_{y}(x_{0},y_{0})\\ -u_{y}(x_{0},y_{0})\end{array}\right),h\to 0.\]The following theorem has thus been proved.  
**Theorem 4**.: _The hyperbolically holomorphic function \(f:D\to\mathfrak{X}_{s}f(\left(\begin{array}{c}x\\ y\end{array}\right))=\left(\begin{array}{c}u(x,y)\\ v(x,y)\end{array}\right)\), satisfies the hyperbolically modified Cauchy-Riemann partial differential equations_  
\[u_{x}=-v_{y}\quad\text{and}\quad v_{x}=-u_{y}.\]  
Consequently, hyperbolically holomorphic functions \(f=(u,v)^{T}\) have the property  
\[u_{xx}-u_{yy}=0.\]