Altzinthings

初一中学生Altzin一年的研究结果
2024.08.23 v3.2
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代数式如下 胆小勿看

$$ \begin{align} &|x| = \sqrt{x^2} \\ &\lfloor x \rfloor = \frac{W\left(i \pi \left(\frac{1}{2} - \frac{2 \arccos\left(\left|\cos\left(\frac{\pi x}{2}\right)\right|\right)}{\pi}\right) (-1)^{-x + \frac{1}{2}}\right) }{i \pi} + x - \frac{1}{2} \\ &\lceil x \rceil = -\lfloor -x \rfloor \\ &f1(x, a, n) = \begin{cases} 0, & x = a \\ n, & x \ne a \end{cases} = \left \lceil \left| \frac{a-x}{|a-x| + 1} \right| \right \rceil n \\ &f2(x, a, n) = \begin{cases} n, & x = a \\ 0, & x \ne a \end{cases} = \left \lfloor 1 - \left| \frac{a-x}{|a-x| + 1} \right| \right \rfloor n \\ &f3(x, a, c, b) = \begin{cases} c, & x = a \\ b, & x \ne a \end{cases} = \left \lceil \left| \frac{a-x}{|a-x| + 1} \right| \right \rceil (b - c) + c \\ &f4(a) = \text{sgn}(a) = \frac{|a|}{a + 1 - f1(a, 0, 1)} \\ &f5(a, x, y, z) = \begin{cases} x, & a < 0 \\ y, & a = 0 \\ z, & a > 0 \end{cases} = \left(\frac{1}{2}z + \frac{1}{2}x - y\right) f4(a)^2 + \frac{z-x}{2} f4(a) + y \end{align} $$