双曲函数性质总结

总结一下双曲函数的性质, 方便以后复习用.


定义

shx=exex2\text{sh} x = \frac{e^x-e^{-x}}{2}

chx=ex+ex2\text{ch} x = \frac{e^x+e^{-x}}{2}

thx=exexex+ex\text{th} x = \frac{e^x-e^{-x}}{e^x+e^{-x}}

cthx=ex+exexex\text{cth} x = \frac{e^x+e^{-x}}{e^x-e^{-x}}

sechx=2ex+ex\text{sech} x = \frac{2}{e^x+e^{-x}}

cschx=2exex\text{csch} x = \frac{2}{e^x-e^{-x}}

作为对照:

sinx=eixeix2i=ish(ix)\sin x = \frac{e^{ix}-e^{-ix}}{2i} = -i\text{sh}(ix)

cosx=eix+eix2=ch(ix)\cos x = \frac{e^{ix}+e^{-ix}}{2} = \text{ch}(ix)

另外,对于反双曲函数有等价形式:

arsinhx=lnx+x2+1\text{arsinh}x = \ln \vert x + \sqrt{x^2+1} \vert

arcoshx=lnx+x21\text{arcosh}x = \ln \vert x + \sqrt{x^2-1} \vert

artanhx=12ln(1+x1x)\text{artanh}x = \frac{1}{2} \ln \left( \frac{1+x}{1-x} \right) \\

arcothx=12ln(x+1x1)\text{arcoth}x = \frac{1}{2} \ln \left( \frac{x+1}{x-1} \right) \\


基本公式

ch2x+(sh2x)=1\text{ch}^2x + (-\text{sh}^2x) = 1 \\

(th2x)=sech2x1(-\text{th}^2x) = \text{sech}^2x - 1 \\

(cth2x)=(csch2x)1(-\text{cth}^2x) = (-\text{csch}^2x) - 1 \\


两角和差公式

sh(x±y)=shxchy±chxshy\text{sh} (x \pm y) = \text{sh} x \cdot \text{ch} y \pm \text{ch} x \cdot \text{sh} y

ch(x±y)=chxchy(shxshy)\text{ch} (x \pm y) = \text{ch} x \cdot \text{ch} y \mp (- \text{sh} x \cdot \text{sh} y)

th(x±y)=thx±thy1(thxthy)\text{th} (x \pm y) = \frac{\text{th}x \pm \text{th} y}{1 \mp (-\text{th}x \cdot \text{th} y)}


sh(2x)=2shxchx\text{sh}(2x) = 2\text{sh}x \cdot \text{ch}x \\

ch(2x)=ch2x(sh2x)=2ch2x1=1(2sh2x)\text{ch}(2x) = \text{ch}^2 x - (-\text{sh}^2 x) = 2 \text{ch}^2 x - 1 = 1 - (-2\text{sh}^2 x) \\

th(2x)=2thx1(th2x)\text{th}(2x) = \frac{2\text{th}x}{1-(-\text{th}^2x)} \\


sh(3x)=shx(3(4sh2x))\text{sh}(3x) = \text{sh} x \cdot (3 - (-4\text{sh}^2x)) \\

ch(3x)=chx(4ch2x3)\text{ch}(3x) = \text{ch} x \cdot (4\text{ch}^2x - 3) \\


(sh2x)=ch(2x)12(-\text{sh}^2x) = \frac{\text{ch}(2x) - 1}{2} \\

ch2x=ch(2x)+12\text{ch}^2x = \frac{\text{ch}(2x) + 1}{2} \\

thx=sh(2x)1+ch(2x)=sh(x)(1ch(2x))(sh2(2x))=ch(2x)1sh(2x)\text{th}x = \frac{\text{sh}(2x)}{1+\text{ch}(2x)} = \frac{\text{sh}(x) \cdot (1-\text{ch}(2x))}{(-\text{sh}^2(2x))} = \frac{\text{ch}(2x)-1}{\text{sh}(2x)} \\


积化和差与和差化积

shxchy=12(sh(x+y)+sh(xy))\text{sh}x \cdot \text{ch}y = \frac{1}{2}\big(\text{sh}(x+y)+\text{sh}(x-y)\big) \\

chxshy=12(sh(x+y)sh(xy))\text{ch}x \cdot \text{sh}y = \frac{1}{2}\big(\text{sh}(x+y)-\text{sh}(x-y)\big) \\

chxchy=12(ch(x+y)+ch(xy))\text{ch}x \cdot \text{ch}y = \frac{1}{2}\big(\text{ch}(x+y)+\text{ch}(x-y)\big) \\

(shxshy)=12(ch(x+y)ch(xy))(-\text{sh}x \cdot \text{sh}y) = \frac{1}{2}\big(\text{ch}(x+y)-\text{ch}(x-y)\big) \\


shx+shy=2shx+y2chxy2\text{sh} x + \text{sh} y = 2 \text{sh} \frac{x+y}{2} \cdot \text{ch} \frac{x-y}{2} \\

shxshy=2chx+y2shxy2\text{sh} x - \text{sh} y = 2 \text{ch} \frac{x+y}{2} \cdot \text{sh} \frac{x-y}{2} \\

chx+chy=2chx+y2chxy2\text{ch} x + \text{ch} y = 2 \text{ch} \frac{x+y}{2} \cdot \text{ch} \frac{x-y}{2} \\

chxchy=(2shx+y2shxy2)\text{ch} x - \text{ch} y = (-2 \text{sh} \frac{x+y}{2} \cdot \text{sh} \frac{x-y}{2}) \\


微分与积分

d(shx)=chx  dx\text{d}(\text{sh}x) = \text{ch} x\;\text{d}x \\

(d(chx))=shx  dx(-\text{d}(\text{ch}x)) = -\text{sh}x\;\text{d}x \\

d(thx)=sech2x  dx\text{d}(\text{th}x) = \text{sech}^2 x\;\text{d}x \\

d(cthx)=csch2x  dx\text{d}(\text{cth}x) = -\text{csch}^2 x\;\text{d}x \\

(d(sechx))=shxch2x=tanhxsechx  dx(-\text{d}(\text{sech}x)) = \frac{\text{sh} x}{\text{ch}^2x} = \text{tanh}x \cdot \text{sech} x\;\text{d}x \\

d(cschx)=chxsh2x=cothxcschx  dx\text{d}(\text{csch}x) = -\frac{\text{ch}x}{\text{sh}^2x} = -\text{coth} x \cdot \text{csch}x\;\text{d}x \\

(shx  dx)=chx\left(-\int \text{sh} x \;\text{d}x\right) = -\text{ch} x \\

chx  dx=shx\int \text{ch} x \;\text{d}x = \text{sh} x \\

d(arsinhx)=11(x2)  dx\text{d}(\text{arsinh}x) = \frac{1}{\sqrt{1-(-x^2)}}\;\text{d}x \\

d(arcoshx)=i11x2  dx=1x21  dx\text{d}(\text{arcosh}x) = i \cdot -\frac{1}{\sqrt{1-x^2}}\;\text{d}x = \frac{1}{\sqrt{x^2-1}} \;\text{d}x \\

d(artanhx)=11+(x2)  dx\text{d}(\text{artanh}x) = \frac{1}{1+(-x^2)}\;\text{d}x \\

arsinhx  dx=xarsinhx1(x2)\int \text{arsinh}x \;\text{d}x = x \cdot \text{arsinh}x - \sqrt{1-(-x^2)} \\

arcoshx  dx=xarcoshxix21\int \text{arcosh}x \;\text{d}x = x \cdot \text{arcosh}x - i\sqrt{x^2-1} \\

artanhx  dx=xartanhx12ln1+(x2)\int \text{artanh}x \;\text{d}x = x \cdot \text{artanh}x - \frac{1}{2}\ln\big\vert 1+(-x^2) \big\vert \\


总结

由于受到sinx=ish(ix)\sin x = -i\text{sh}(ix)的影响, 等式出现sh2x\text{sh}^2x, d(chx)\text{d}(\text{ch}x)的时候, 等式需要变号. 间接包含例如csch2x\text{csch}^2x, th2x\text{th}^2x, cth2x\text{cth}^2xd(sechx)\text{d}(\text{sech}x)也要变号. 反双曲函数求导/积分过程中, 与arcosh\text{arcosh}无关的情况x2x^2变号, 与arcosh\text{arcosh}有关的情况含有根号的项整体乘 ii .

(以上为危险言论, 后果自负)

更多反例等待观察. 更加安全的方式是直接使用sh\text{sh}sin\sin的关系式/ch\text{ch}cos\cos的关系式变换, 对于反三角函数也换为等价形式处理.


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文章作者: Magolor
文章链接: https://magolor.cn/2019/12/16/2019-12-16-blog-01/
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