基于反三角函数和反双纽线函数类比的叶函数整根加法公式
Addition Formulas of Leaf Functions According to Integral Root of Polynomial Based on Analogies of Inverse Trigonometric Functions and Inverse Lemniscate Functions
摘要：
函数r（t）对变量t的二阶导数等于r（t）的2n-1次幂的函数的-n倍;利用这个定义，可以构造一个常微分方程。通过数值求解常微分方程来创建具有作为变量t的水平轴和作为变量r（t）的垂直轴的图。这些图显示了几个规则波，其具有特定的周期性和波形，取决于自然数n。在这项研究中，满足常微分方程的函数作为叶函数。对于n = 1，叶函数变成三角函数。对于n = 2，叶函数变成双系椭圆函数。这些函数涉及一个附加定理。本文在n = 1和n = 2的附加定理的基础上，给出了n = 3的倍角和附加定理。
英文摘要：
The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal axis as the variable t and the vertical axis as the variable r(t) are created by numerically solving the ordinary differential equation. These graphs show several regular waves with a specific periodicity and waveform depending on the natural number n. In this study, the functions that satisfy the ordinary differential equation are presented as the leaf functions. For n = 1, the leaf function become a trigonometric function. For n = 2, the leaf function becomes a lemniscatic elliptic function. These functions involve an additional theorem. In this paper, based on the additional theorem for n = 1 and n = 2, the double angle and additional theorem for n = 3 are presented.
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