简单根式扩张上的并行积分 III：以特殊函数表示的反导数
Parallel Integration over Simple Radical Extensions III: Antiderivatives in Terms of Special Functions
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中文总结 AI 辅助
本文扩展并行积分方法至特殊函数反导数，覆盖不完全伽马函数、椭圆积分等，通过核与线性系统引入，证明仅在必要时引入特殊函数，并给出严格模式证书与偏答案。
中文摘要 AI 辅助
我们将第二部分中混合塔的并行积分方法从初等反导数扩展到特殊函数反导数。所覆盖的类别是自变量为有理数 $s$ 的不完全伽马函数 $\Gamma(s,\cdot)$，它包含 $\Ei$、$\li$、$\Si$、$\Ci$、$\erf$ 和菲涅耳积分，以及椭圆积分 $F$、$E$、$\Pi$。每个特殊函数通过一个“核”进入：塔中一个已知元素，其反导数即为该函数。该核要么由留数确定，要么作为单线性系统的一列添加，且不求解 Risch 微分方程。我们证明了核可能具有极点的位置；$\erf$ 核位于根式塔的次临界窗口内。第二部分的分母理论、次数界和证书得以沿用，新的判据决定了第二部分留给猜测的多数位置。椭圆积分由根式承载，非挠留数因子成为第三类项。我们证明仅在必要时才引入特殊函数。初等答案原样返回，在严格模式下，特殊函数附带被积函数无初等积分的证书。当找不到完整答案时，返回带有简化余项的偏答案。所有示例均由 SymPy 实现计算，并通过微分验证。
英文摘要
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.
