Parallel Integration over Simple Radical Extensions III: Antiderivatives in Terms of Special Functions

Abstract

We extend the parallel integration method for mixed towers of Part~II from elementary antiderivatives to antiderivatives in special functions. The class covered is the incomplete gamma function $Γ(s,\cdot)$ at rational $s$, which contains $\Ei$, $\li$, $\Si$, $\Ci$, $\erf$ and the Fresnel integrals, together with the elliptic integrals $F$, $E$, $Π$. Each special function enters through a \emph{kernel}: a known element of the tower whose antiderivative is that function. It is either fixed by residues or added as one more column of the single linear system, and no Risch differential equation is solved. We prove where kernels can have poles; the $\erf$ kernels live in the sub-critical window of radical towers. The denominator theory, degree bounds and certificates of Part~II carry over, and new criteria decide most places that Part~II leaves to a guess. Elliptic integrals are carried by the radical, and a non-torsion residue divisor becomes a third-kind term. We prove that special functions are introduced only when necessary. Elementary answers are returned unchanged, and in strict mode a special function comes with a certificate that the integrand has no elementary integral. When no complete answer is found, a partial answer with a reduced remainder is returned. All examples are computed by a SymPy implementation and verified by differentiation.