  We consider the bending angle of the trajectory of a photon incident from and
deflected to infinity around a Reissner-Nordstr\"om black hole. We treat the
bending angle as a function of the squared reciprocal of the impact parameter
and the squared electric charge of the background normalized by the mass of the
black hole. It is shown that the bending angle satisfies a system of two
inhomogeneous linear partial differential equations with polynomial
coefficients. This system can be understood as an isomonodromic deformation of
the inhomogeneous Picard-Fuchs equation satisfied by the bending angle in the
Schwarzschild spacetime, where the deformation parameter is identified as the
background electric charge. Furthermore, the integrability condition for these
equations is found to be a specific type of the Painlev\'e VI equation that
allows an algebraic solution. We solve the differential equations both at the
weak and strong deflection limits. In the weak deflection limit, the bending
angle is expressed as a power series expansion in terms of the squared
reciprocal of the impact parameter and we obtain the explicit full-order
expression for the coefficients. In the strong deflection limit, we obtain the
asymptotic form of the bending angle that consists of the divergent logarithmic
term and the finite O(1) term supplemented by linear recurrence relations which
enable us to straightforwardly derive higher order coefficients. In deriving
these results, the isomonodromic property of the differential equations plays
an important role. Lastly, we briefly discuss the applicability of our method
to other types of spacetimes such as a spinning black hole.
