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We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called $$\alpha $$-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric $$\begin{aligned} ds_{\alpha }^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\alpha }} \end{aligned}$$in the upper half space $${\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}$$. If $$\alpha =2$$, the metric is the hyperbolic metric of the Poincare upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function $$x^{m}\,(m\in {\mathbb {Z}})$$, is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental $$\alpha $$-hyperbolic harmonic functions, depending only on the hyperbolic distance and $$x_{3}$$, we verify a Cauchy type integral formula for conjugate gradient of $$\alpha $$-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued $$\alpha $$-hypermonogenic in the Clifford algebra $${{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}$$.

Hyperbolic Function Theory in the Skew-Field of Quaternions