  The timelike geodesic equations resulting from the Kerr gravitational metric
element are derived and solved exactly including the contribution from the
cosmological constant. The geodesic equations are derived, by solving the
Hamilton-Jacobi partial differential equation by separation of variables. The
solutions can be applied in the investigation of the motion of a test particle
in the Kerr and Kerr-(anti) de Sitter gravitational fields. In particular, we
apply the exact solutions of the timelike geodesics i) to the precise
calculation of dragging (Lense-Thirring effect) of a satellite's spherical
polar orbit in the gravitational field of Earth assuming Kerr geometry, ii)
assuming the galactic centre is a rotating black hole we calculate the precise
dragging of a stellar polar orbit aroung the galactic centre for various values
of the Kerr parameter including those supported by recent observations. The
exact solution of non-spherical geodesics in Kerr geometry is obtained by using
the transformation theory of elliptic functions. The exact solution of
spherical polar geodesics with a nonzero cosmological constant can be expressed
in terms of Abelian modular theta functions that solve the corresponding
Jacobi's inversion problem.
