Taylor’s Series

Definition

$$ \begin{gather} \begin{aligned} f(x - a) &= \sum_{ n = 0 }^{ \infin } \frac{ f^{(n)} (a)}{ n! }(x -a)^n \\ f(x) &= \sum_{ n = 0 }^{ \infin } \frac{ f^{(n)} (0)}{ n! } x^n \end{aligned} \end{gather} $$

常见Taylor’s Series

$$ \begin{gather} \begin{aligned} (1+x)^\alpha &= 1 + \sum_{ n = 1 }^{ \infin } \frac{ \alpha ( \alpha -1)…( \alpha - n + 1) }{ n ! } x^n \\ &= 1 + \alpha x + \frac{ \alpha( \alpha -1) }{ 2! } x^2 + …, x \in (-1,1) \end{aligned} \end{gather} $$

三角函数

双曲函数 & 三角函数

$$ \begin{gather} \begin{aligned} \sin(i x) &= i \sinh(x) \\ \cos(i x) &= \cosh (x) \end{aligned} \end{gather} $$

正切函数

$$ \begin{gather} \begin{aligned} \tan ( \alpha + \beta) &= \frac{ \tan \alpha + \tan \beta}{ 1 - \tan \alpha \tan \beta } \\ \tan (ix) &= i \tanh x \end{aligned} \end{gather} $$

双曲函数

$$ \begin{gather} \begin{aligned} \sinh x &= \frac{ e^x - e^{-x} }{ 2 }) \\ \cosh x &= \frac{ e^x + e^{-x} }{ 2 } \\ \tanh x &= \frac{ e^{2x} - 1 }{ e^{2x} + 1} \end{aligned} \end{gather} $$

反三角函数

$$ \begin{gather} \begin{aligned} \tan^{-1} (A) + \tan^{-1} (B) &= \tan^{-1} \Big( \frac{ A + B }{ 1 - AB } \Big) \end{aligned} \end{gather} $$