A hyperbolic parametrization of lepton mixing for small $U_{e3}$

We use hyperbolic functions to parametrize lepton mixing matrix using only one input parameter $\phi$. This matrix has three mixing angles as outputs therefore giving two predictions. In particular it predicts $U_{e3}=0$ besides predicting correct solar and atmospheric mixing. We confine us to real $\phi$. For complex $\phi$ mixing matrix is no longer unitary. Next we accentuate this unitary mixing matrix with an additional small parameter $\delta$ while keeping unitarity of the matrix exact. When this second parameter is included, with this framework, one can handle small but non-zero $U_{e3}$. In the second case from two input parameters one obtains three mixing angles thus one prediction.