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In this work, we aim at solving the Teukolsky equations for a fermion withmass $m_e\neq 0$ in the presence of a rotating black hole with mass $M$. Weconsider two different regimes: $\tilde{m}_e= M^{-1} m_e\ll 1$ and $a\omega \ll1$; $\tilde{m}_e\ll 1$ and $a\omega \gtrsim 1$. We treat each of these tworegimes in different ways: we use a perturbative approach for the first,similar to the \emph{usual} one employed for spin 0, 1, 2 and mass-less 1/2fields, but with two small parameters (a$\omega$ and $\tilde{m}_e$); as weshall see, the second can be treated with a semi-analytical approach. In aforthcoming paper we shall study the remaining two cases in which $\tilde{m}_e\gtrsim 1$, while $a\omega \ll 1$ or $a\omega \gtrsim 1$. The regime with$\tilde{m}_e \ll 1$, but $a\omega\gtrsim 1$ is probably the most interestingfrom the astrophysics point of view, but this last two cases might be of someinterest for the study of the interaction of fermions with very small blackholes, which may be formed, for example, in the last stages of the Hawkingevaporation.

Perturbative and semi-analytical solutions to Teukolsky equations for massive fermions