  We consider the construction of the fundamental function and Abelian
differentials of the third kind on a plane algebraic curve over the field of
complex numbers that has no singular points. The algorithm for constructing
differentials of the third kind is described in Weierstrass's Lectures. The
article discusses its implementation in the Sage computer algebra system. The
specificity of this algorithm, as well as the very concept of the differential
of the third kind, implies the use of not only rational numbers, but also
algebraic ones, even when the equation of the curve has integer coefficients.
Sage has a built-in algebraic number field tool that allows implementing
Weierstrass's algorithm almost verbatim. The simplest example of an elliptic
curve shows that it requires too many resources, going far beyond the
capabilities of an office computer. Then the symmetrization of the method is
proposed and implemented, which solves the problem and allows significant
economy of resources. The algorithm for constructing a differential of the
third kind is used to find the value of the fundamental function according to
the duality principle. Examples explored in the Sage system are provided.
