画板前，墨汁还没干透，我就想，那组对数公式是不是该换个说法？别整那些“极限过程”、“定义域边界”的学术腔调，咱就把它当成手里的算盘，拨动两下，看看能不能顺嘴蹦出来。 反正、反正、反正…… 在脑子里疯狂蹦迪的时候，我就突然灵光一闪：能不能把那个阿贝尔函数给弄成个对数？ 你看，反正符号 `arccos` 和 `arcsin`，啥？不就是 $arccos x = arctan(frac{sqrt{1-x^2}}{x})$ 吗？ 这玩意儿那会儿得换个角度凑，目前直接套个反正弦进去，不就出了个反正切？ 这就好比把拧螺丝的手，从侧面转到正面，姿势直接优雅了。 同理，`arccsc` 和 `arcsec`，这两个我也琢磨两天了，反正 $arccsc x = text{arccot}(frac{1}{x})$，`arcsec x` 对应 $text{arccos}(1/x)$，这两个公式要是抄下来，直接代入现有的对数工具，秒杀鸡皮疙瘩。 还有那个 `arccoth`，$x = text{arccoth}(t)$ 这种反向关系，直接写成 $text{arctanh}(1/t)$ 也行，反正都是 $text{arctanh}$ 的亲戚。 最终那个反正切，`arctan x`，公式直接就是 $frac{pi}{2} - text{arccot } x$，要么直接用 $text{arctan}(1/x)$ 搞定，这就够了。 这时候再回头看看那些看起来像天书一样的公式，比如洛必达那种极限，要么反三角函数的微分公式，是不是也该换种心情？ 别逼自己在那儿背一堆导数表的玩意儿了，那些玩意儿实际上是积分表里的倒过来。 想求反正弦的导数？别费劲了，直接拿 `arcsin x` 去加一个 $frac{1}{sqrt{1-x^2}}$，这不就是积分凑出来的结局嘛。 想求积分？直接拿 `arctan x` 去乘一个 $dx$，这不就是幂函数的积分了？ 我们那会儿学微积分，总认定函数和积分是两码事，目前想想，它们不过是同一枚硬币的两面，翻过来就是对的，反着来就是对的。 在这种“废话文学”的视角下，大量推导过程显得特别无聊，特别是那些为了凑出来的常数项，反正不管凑出多少，反正都是个常数。 比如求 $int text{sech} x dx$，那是 $arctan(sinh x)$ 啊，但这玩意儿里藏着个常数 $pi/4$，这常数到底是哪位带来的？ 不管怎么着，反正它就是个常数，跟解题的大逻辑关系不大，跟计算结局的大小无涉。 故此，还不如在这些细节里纠缠，不如把这层窗户纸捅破，直接看结局。 反正 $x in (-1, 1)$ 时，反余弦就是那个 $arctan$ 的变体，反正 $x in (-infty, -1) cup (1, infty)$ 时，反余切的那个变体，反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反正反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