Expressions for neutrino wave functions and transition probabilities at three neutrino transitions (oscillations) in vacuum

Three neutrino vacuum transitions and oscillations in the general case were considered and expressions for neutrino wave functions in three cases: with $CP$ violation, without $CP$ violation and the case when direct $\nu_e\leftrightarrow\nu_\tau$ transitions are absent $\beta(\theta_{13})=0$ (some works indicate on this possibility) were obtained. Also were computed transition probabilities for the case of $CP$ conservation and when direct $\nu_e\leftrightarrow\nu_\tau$ transitions are absent i.e., $\beta(\theta_{13})=0$. It was found out that the probability of $\nu_e\leftrightarrow\nu_e$ neutrino transitions is positive defined value only if the angle of $\nu_e,\nu_\tau$ mixing $\beta<15^\circ\div17^\circ$.