AI 中文总结
本文提出一种统一除子比较方法，通过比较零点极点多重集并利用固定魏尔斯特拉斯规范化，以四步流程证明亚纯函数恒等式，涵盖多个经典函数族，实现方法论综合。
AI 中文摘要
我们提出了一种通过比较完整的零点和极点多重集来证明亚纯函数恒等式的统一方法。一个固定的环形魏尔斯特拉斯规范化将每个非零亚纯函数唯一地表示为 \\[ q_t=e^{h_t}\frac{p_{Z_t}}{p_{P_t}},\qquad t=([h_t],Z_t,P_t),\qquad [h_t]\in\mathcal O(\mathbb C)/(2\pi i\mathbb Z). \\] 十三个主要比较及其特化和推论使用相同的四个步骤：表示两侧、比较它们的除子、应用除子比较定理（DCT），并规范化 $h_s-h_t$。一个小型库将经典的有限阶、奇偶性、递推和格刚性论证打包以供重复使用。例子涵盖正弦、伽马、巴恩斯 $G$、贝塞尔、完备化zeta、theta和椭圆函数。它们的广度展示了在这些函数族中重复使用一个证明结构；族特定的假设和标量计算保持显式。贡献在于方法论的综合：在可能的情况下给出更短的证明，否则对经典论证进行模块化组织。固定因子提供公共坐标，而非新的唯一性假设。从历史上看，这一观点受到十九世纪黎曼对复函数理论的全局几何方法与魏尔斯特拉斯对函数的解析构造之间对比的启发；本方法有意将全局除子数据与固定魏尔斯特拉斯因子相结合。附录独立于证明中使用的普通运算，发展了相关的环和域表示。
英文摘要
We present a uniform method for proving meromorphic function identities by comparing complete zero and pole multisets. A fixed annular Weierstrass normalization represents every nonzero meromorphic function uniquely as \[ q_t=e^{h_t}\frac{p_{Z_t}}{p_{P_t}},\qquad t=([h_t],Z_t,P_t),\qquad [h_t]\in\mathcal O(\mathbb C)/(2πi\mathbb Z). \] Thirteen principal comparisons, together with their specializations and consequences, use the same four steps: represent the two sides, compare their divisors, apply a divisor-comparison theorem (DCT), and normalize $h_s-h_t$. A small library packages classical finite-order, parity, recurrence, and lattice-rigidity arguments for repeated use. The examples span sine, Gamma, Barnes $G$, Bessel, completed zeta, theta, and elliptic functions. Their breadth demonstrates reuse of one proof structure across these families; the family-specific hypotheses and scalar calculations remain explicit. The contribution is a methodological synthesis: shorter proofs where available, and otherwise a modular organization of classical arguments. The fixed factors supply common coordinates, rather than a new uniqueness hypothesis. Historically, the viewpoint is motivated by the nineteenth-century contrast between Riemann's global geometric approach to complex function theory and Weierstrass's analytic construction of functions; the present method deliberately combines global divisor data with fixed Weierstrass factors. An appendix develops the associated ring and field presentations, separately from the ordinary operations used in the proofs.
