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We study transcendental meromorphic functions with essential singularities onRiemann surfaces. Every function $\Psi_X$ has associated a complex vector field$X$. In the converse direction, vector fields $X$ provide single valued ormultivalued functions $\Psi_X$. Our goal is to understand the relationshipbetween the analytical properties of $\Psi_X$, the singularities of its inverse$\Psi_X^{-1}$ and the geometric behavior of $X$. As first result, by examiningthe containment properties of the neighborhoods of the singularities of$\Psi_X^{-1}$, we characterize when a singularity of $\Psi_X^{-1}$ over asingular value $a$, is either algebraic, logarithmic or non logarithmic.Secondly, to make precise the cooperative aspects between analysis andgeometry, we present the systematic study of two holomorphic families oftranscendental functions with an essential singularity at infinity, as well assome sporadic examples. As third stage, we study the incompleteness of thetrajectories of the associated vector field $X$ with essential singularities ona Riemann surface $M_{\mathfrak g}$ of genus ${\mathfrak g}$. As anapplication, we provide conditions under which there exists an infinite numberof (real) incomplete trajectories of $X$ localized at the essentialsingularities. Furthermore, removing the incomplete trajectories decomposes theRiemann surface into real flow invariant canonical pieces.

Geometry of transcendental singularities of complex analytic functions and vector fields