2.9.3 有关双曲函数的重要公式

双曲函数之间存在着与三角函数之间类似的关系, 利用双曲函数的定义可以直接证明接下来的公式成立. 另外,由 $\left(2.199\right)\sim \left(2.206\right)$ ,考虑到当自变量为复数时这些函数的定义和关系, 可以利用已知的三角函数公式计算它们.

2.9.3.1 单变量双曲函数

$$\begin{array}{}\text{(2.172)}& {\cosh }^{2}x-{\sinh }^{2}x=1,\end{array}$$$$\begin{array}{}\text{(2.173)}& {\coth }^{2}x-{\mathrm{csch}}^{2}x=1,\end{array}$$$$\begin{array}{}\text{(2.174)}& {\mathrm{sech}}^{2}x+{\tanh }^{2}x=1,\end{array}$$$$\begin{array}{}\text{(2.175)}& \tanh x\cdot \coth x=1,\end{array}$$$$\begin{array}{}\text{(2.176)}& \frac{\sinh x}{\cosh x}=\tanh x\end{array}$$$$\begin{array}{}\text{(2.177)}& \frac{\cosh x}{\sinh x}=\coth x\end{array}$$

2.9.3.2 某一双曲函数用具有相同自变量的另一个双曲函数的 表示

相应公式见表 2.7.

表 $2.7x>0$ 时,具有相同自变量的两双曲函数间的关系

	$\sinh x$	$\cosh x$	$\tanh x$	$\coth x$
$\sinh x$		$\sqrt{{\cosh }^{2}x-1}$	$\frac{\tanh x}{\sqrt{1-{\tanh }^{2}x}}$	$\frac{1}{\sqrt{{\coth }^{2}x-1}}$
$\cosh x$	$\sqrt{{\sinh }^{2}x+1}$		$\frac{1}{\sqrt{1-{\tanh }^{2}x}}$	$\frac{\coth x}{\sqrt{{\coth }^{2}x-1}}$
$\tanh x$	$\frac{\sinh x}{\sqrt{{\sinh }^{2}x+1}}$	$\frac{\sqrt{{\cosh }^{2}x-1}}{\cosh x}$		$\frac{1}{\coth x}$
$\coth x$	$\frac{\sqrt{{\sinh }^{2}x+1}}{\sinh x}$	$\frac{\cosh x}{\sqrt{{\cosh }^{2}x-1}}$	$\frac{1}{\tanh x}$

2.9.3.3 负角公式

$$\begin{array}{}\text{(2.178)}& \sinh \left(-x\right)=-\sinh x,\end{array}$$$$\begin{array}{}\text{(2.179)}& \tanh \left(-x\right)=-\tanh x,\end{array}$$$$\begin{array}{}\text{(2.180)}& \cosh \left(-x\right)=\cosh x,\end{array}$$$$\begin{array}{}\text{(2.181)}& \coth \left(-x\right)=-\coth x.\end{array}$$

2.9.3.4 两自变量和与差的双曲函数 (加法定理)

$$\begin{array}{}\text{(2.182)}& \sinh \left(x\pm y\right)=\sinh x\cosh y\pm \cosh x\sinh y,\end{array}$$$$\begin{array}{}\text{(2.183)}& \cosh \left(x\pm y\right)=\cosh x\cosh y\pm \sinh x\sinh y,\end{array}$$$$\begin{array}{}\text{(2.184)}& \tanh \left(x\pm y\right)=\frac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y},\end{array}$$$$\begin{array}{}\text{(2.185)}& \coth \left(x\pm y\right)=\frac{1\pm \coth x\coth y}{\coth x\pm \coth y}.\end{array}$$

2.9.3.5 倍角的双曲函数

$$\begin{array}{}\text{(2.186)}& \sinh 2x=2\sinh x\cosh x,\end{array}$$$$\begin{array}{}\text{(2.187)}& \cosh 2x={\sinh }^{2}x+{\cosh }^{2}x,\end{array}$$$$\begin{array}{}\text{(2.188)}& \tanh 2x=\frac{2\tanh x}{1+{\tanh }^{2}x},\end{array}$$$$\begin{array}{}\text{(2.189)}& \coth 2x=\frac{1+{\coth }^{2}x}{2\coth x}.\end{array}$$

2.9.3.6 双曲函数的棣莫弗公式

$$\begin{array}{}\text{(2.190)}& {(\cosh x\pm \sinh x)}^n={\left({\mathrm{e}}^{\pm x}\right)}^n={\mathrm{e}}^{\pm nx}=\cosh nx\pm \sinh nx.\end{array}$$

2.9.3.7 半角的双曲函数

$$\begin{array}{}\text{(2.191)}& \sinh \frac{x}{2}=\pm \sqrt{\frac{1}{2}\left(\cosh x-1\right)},\end{array}$$$$\begin{array}{}\text{(2.192)}& \cosh \frac{x}{2}=\sqrt{\frac{1}{2}\left(\cosh x+1\right)},\end{array}$$

当 $x>0$ 时,(2.191) 中平方根的符号为正; 当 $x<0$ 时,符号为负.

$$\begin{array}{}\text{(2.193)}& \tanh \frac{x}{2}=\frac{\cosh x-1}{\sinh x}=\frac{\sinh x}{\cosh x+1},\end{array}$$$$\begin{array}{}\text{(2.194)}& \coth \frac{x}{2}=\frac{\sinh x}{\cosh x-1}=\frac{\cosh x+1}{\sinh x}.\end{array}$$

2.9.3.8 双曲函数的和与差

$$\begin{array}{}\text{(2.195)}& \sinh x\pm \sinh y=2\sinh \frac{x\pm y}{2}\cosh \frac{x\mp y}{2},\end{array}$$$$\begin{array}{}\text{(2.196)}& \cosh x+\cosh y=2\cosh \frac{x+y}{2}\cosh \frac{x-y}{2},\end{array}$$$$\begin{array}{}\text{(2.197)}& \cosh x-\cosh y=2\sinh \frac{x+y}{2}\sinh \frac{x-y}{2},\end{array}$$$$\begin{array}{}\text{(2.198)}& \tanh x\pm \tanh y=\frac{\sinh \left(x\pm y\right)}{\cosh x\cosh y}.\end{array}$$

2.9.3.9 复角 $z$ 的双曲函数与三角函数间的关系

$$\begin{array}{}\text{(2.199)}& \sin z=-\mathrm{i}\sinh \mathrm{i}z,\end{array}$$$$\begin{array}{}\text{(2.200)}& \cos z=\cosh \mathrm{i}z\end{array}$$$$\begin{array}{}\text{(2.201)}& \tan z=-\mathrm{i}\tanh \mathrm{i}z,\end{array}$$$$\begin{array}{}\text{(2.202)}& \cot z=\mathrm{i}\coth \mathrm{i}z\end{array}$$$$\begin{array}{}\text{(2.203)}& \sinh z=-\mathrm{i}\sin iz,\end{array}$$$$\begin{array}{}\text{(2.204)}& \cosh z=\cos \mathrm{i}z\end{array}$$$$\begin{array}{}\text{(2.205)}& \tanh z=-\mathrm{i}\tan iz,\end{array}$$$$\begin{array}{}\text{(2.206)}& \coth z=\mathrm{i}\cot \mathrm{i}z.\end{array}$$

通过把 $\sin \alpha$ 代换成 $\mathrm{i}\sinh x$ ,把 $\cos \alpha$ 代换成 $\cosh x$ ,利用相应的三角函数关系,可以得到关于 $x$ 或 $ax$ 但不能得到 $ax+b$ 的双曲函数间的关系.

$◼\mathbf{A}$: ${\cos }^2\alpha +{\sin }^2\alpha =1,{\cosh }^{2}x+{\mathrm{i}}^2{\sinh }^{2}x=1$ 或 ${\cosh }^{2}x-{\sinh }^{2}x=1$ .

$◼\mathbf{B}$: $\sin 2\alpha =2\sin \alpha \cos \alpha ,\mathrm{i}\sinh 2x=2\mathrm{i}\sinh x\cosh x$ 或

$\sinh 2x=2\sinh x\cosh x.$