Гиперболические функции
Определение гиперболических функций
\(\mathrm{sh}{x}=\displaystyle\frac{e^x-e^{-x}}{2}\)
\(\mathrm{ch}{x}=\displaystyle\frac{e^x+e^{-x}}{2}\)
\(\mathrm{th}{x}=\displaystyle\frac{\mathrm{sh}{x}}{\mathrm{ch}{x}}=\displaystyle\frac{e^x-e^{-x}}{e^x+e^{-x}}\)
\(\mathrm{cth}{x}=\displaystyle\frac{\mathrm{ch}{x}}{\mathrm{sh}{x}}=\displaystyle\frac{e^x+e^{-x}}{e^x-e^{-x}}\)
Основные тождества
\(\mathrm{ch}^2{x}-\mathrm{sh}^2{x}=1\)
\(\mathrm{th}{x}\cdot\mathrm{cth}{x}=1\)
\(1-\mathrm{th}^2{x}=\displaystyle\frac{1}{\mathrm{ch}^2{x}}\)
\(\mathrm{cth}^2{x}-1=\displaystyle\frac{1}{\mathrm{sh}^2{x}}\)
Формулы сложения
\(\mathrm{sh}(x+y)=\mathrm{sh}x\cdot\mathrm{ch}y+\mathrm{ch}x\cdot\mathrm{sh}y\)
\(\mathrm{sh}(x-y)=\mathrm{sh}x\cdot\mathrm{ch}y-\mathrm{ch}x\cdot\mathrm{sh}y\)
\(\mathrm{ch}(x+y)=\mathrm{ch}x\cdot\mathrm{ch}y+\mathrm{sh}x\cdot\mathrm{sh}y\)
\(\mathrm{ch}(x-y)=\mathrm{ch}x\cdot\mathrm{ch}y-\mathrm{sh}x\cdot\mathrm{sh}y\)
\(\mathrm{th}(x+y)=\displaystyle\frac{\mathrm{th}x+\mathrm{th}y}{1+\mathrm{th}x\cdot\mathrm{th}y}\)
\(\mathrm{th}(x-y)=\displaystyle\frac{\mathrm{th}x-\mathrm{th}y}{1-\mathrm{th}x\cdot\mathrm{th}y}\)
Сумма и разность в произведение
\(\mathrm{sh}x+\mathrm{sh}y=2\mathrm{sh}\displaystyle\frac{x+y}{2}\mathrm{ch}\displaystyle\frac{x-y}{2}\)
\(\mathrm{sh}x-\mathrm{sh}y=2\mathrm{sh}\displaystyle\frac{x+y}{2}\mathrm{ch}\displaystyle\frac{x+y}{2}\)
\(\mathrm{ch}x+\mathrm{ch}y=2\mathrm{ch}\displaystyle\frac{x+y}{2}\mathrm{ch}\displaystyle\frac{x-y}{2}\)
\(\mathrm{ch}x-\mathrm{ch}y=2\mathrm{sh}\displaystyle\frac{x+y}{2}\mathrm{sh}\displaystyle\frac{x-y}{2}\)
\(\mathrm{th}x+\mathrm{th}y=\displaystyle\frac{\mathrm{sh}(x+y)}{\mathrm{ch}x\cdot\mathrm{ch}y}\)
\(\mathrm{th}x-\mathrm{th}y=\displaystyle\frac{\mathrm{sh}(x-y)}{\mathrm{ch}x\cdot\mathrm{ch}y}\)
Произведение в сумму
\(\mathrm{sh}x\cdot\mathrm{ch}y=\displaystyle\frac{1}{2}(\mathrm{sh}(x+y)+\mathrm{sh}(x-y))\)
\(\mathrm{ch}x\cdot\mathrm{ch}y=\displaystyle\frac{1}{2}(\mathrm{ch}(x+y)+\mathrm{ch}(x-y))\)
\(\mathrm{sh}x\cdot\mathrm{sh}y=\displaystyle\frac{1}{2}(\mathrm{ch}(x+y)-\mathrm{ch}(x-y))\)
Формулы двойного аргумента
\(\mathrm{sh}2x=2\mathrm{sh}x\cdot\mathrm{ch}x\)
\(\mathrm{ch}2x=\mathrm{sh}^2x+\mathrm{ch}^2x\)
\(\mathrm{ch}2x=1+2\mathrm{sh}^2x\)
\(\mathrm{ch}2x=2\mathrm{ch}^2x-1\)
\(\mathrm{th}2x=\displaystyle\frac{2\mathrm{th}x}{1+\mathrm{th}^2x}\)
Формулы половинного аргумента
\(\mathrm{sh}x=\displaystyle\frac{2\mathrm{th}\displaystyle\frac{x}{2}}{1-\mathrm{th}^2\displaystyle\frac{x}{2}}\)
\(\mathrm{ch}x=\displaystyle\frac{1+\mathrm{th}^2\displaystyle\frac{x}{2}}{1-\mathrm{th}^2\displaystyle\frac{x}{2}}\)
\(\mathrm{th}x=\displaystyle\frac{2\mathrm{th}\displaystyle\frac{x}{2}}{1+\mathrm{th}^2\displaystyle\frac{x}{2}}\)
Формулы понижения степени
\(\mathrm{sh}^2{x}=\displaystyle\frac{1}{2}(\mathrm{ch}2x-1)\)
\(\mathrm{ch}^2x=\displaystyle\frac{1}{2}(\mathrm{ch}2x+1)\)
\(\mathrm{th}^2x=\displaystyle\frac{\mathrm{ch}2x-1}{\mathrm{ch}2x+1}\)
\((\mathrm{sh}x+\mathrm{ch}x)^n=\mathrm{sh}nx+\mathrm{ch}nx\)