求解多项式方程中幂级数合并的形式体系：主级数与主数
A Formalism for Merging Power Series in Solving Multinomial Equations: Master Series and Master Numbers
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中文总结 AI 辅助
本文提出主级数与主数形式体系，统一表示多项式方程各类解，并给出收敛半径外共轭级数选取准则，实现级数合并，应用于气体动力学与等离子体物理。
中文摘要 AI 辅助
本文证明，具有任意复指数 a、b（情形 m = 1）的规范方程 y^a = 1 + a*x*y^b 的所有解，及其极限情形——在 b = 0 时的二项式根 y = (1 + a*x)^(1/a)，在 a = 0、b = 0 时的指数函数 y = e^x，以及在 a = 0 时的 Lambert 方程 y = e^(x*y^b)——以及这些解的对数（情形 m = 0），均可由单一幂级数主级数 M(m; a; b; x) = m + x + ((m - a + 2b)/2!) x^2 + ((m - a + 3b)(m - 2a + 3b)/3!) x^3 + ... 表示。对于收敛半径之外的多值解，本文提出了一种几何准则来选取共轭级数（即相同的主级数，但参数集不同，为 M(m; a'; b'; x')），从而确保在复平面上的分支连续性。这种基于主数的幂级数表示不仅允许在 gamma 函数表示失去意义的退化情形下计算系数，而且为主级数的合并操作提供了一种透明的解析算法。该方法在气体动力学和等离子体物理问题中得到了说明，结果表明，物理上可实现的状态需要在基本展开的收敛半径之外应用共轭级数。
英文摘要
It is shown that all solutions of the canonical equation y^a = 1 + a*x*y^b with arbitrary complex exponents a, b (case m = 1) and of its limiting cases - the binomial root y = (1 + a*x)^(1/a) at b = 0, the exponential y = e^x at a = 0, b = 0, and the Lambert equation y = e^(x*y^b) at a = 0 - as well as the logarithms of these solutions (case m = 0), are expressed by a single power master series M(m; a; b; x) = m + x + ((m - a + 2b)/2!) x^2 + ((m - a + 3b)(m - 2a + 3b)/3!) x^3 + ... For multivalued solutions beyond the radius of convergence, a geometric criterion for selecting the conjugate series (the same master series, but with a different set of parameters M(m; a'; b'; x')) is proposed, ensuring branch continuity on the complex plane. This representation of power series in terms of master numbers not only allows one to compute coefficients in degenerate cases where the gamma-function representation loses meaning, but also provides a transparent analytical algorithm for the merge operation on master series. The method is illustrated on problems in gas dynamics and plasma physics, where it is demonstrated that physically realizable states require the application of conjugate series beyond the radius of convergence of the basic expansion.
发表机构
- Municipal Budgetary Institution, Iglino（伊格利诺市政预算机构）
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