$\newcommand{\ddx}[1]{\frac{d}{dx}\left[#1\right]}$

Hyperbolic Functions

Definitions, Theorems, and Lemmas

Complex Numbers and Trigonometry

Relationship to trigonometry $$\begin{aligned}\cos(x) & =\Re(e^{ix})=\dfrac{e^{ix}+e^{-ix}}{2} & \sin(x) & =\Im(e^{ix})=\dfrac{e^{ix}-e^{-ix}}{2i}\\ \cos(iy) & =\dfrac{e^{y}+e^{-y}}{2}=\cosh(y) & \sin(iy) & =\dfrac{e^{y}-e^{-y}}{2i}=i\sinh(y) \end{aligned}$$

The Sum of Two Angles

The Sum of Two Angles $$\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$$ $$\sin(\alpha+\beta)=\cos(\alpha)\sin(\beta)+\cos(\beta)\sin(\alpha)$$

Proof. Sum of Angles $$\begin{aligned} \cos(\alpha+\beta) & =\Re\left(e^{i(\alpha+\beta)}\right) & i & =\sqrt{-1}\\ & =\Re\left(e^{i\alpha}e^{i\beta}\right)\\ & =\Re\left((\cos(\alpha)+i\sin(\alpha)(\cos(\beta)+i\sin(\beta)\right) & i^{2} & =-1\\ & =\Re\left(\cos(\alpha)\cos(\beta)+i\left(\cos(\alpha)\sin(\beta)+\sin(\alpha)\cos(\beta)\right)+i^{2}\sin(\alpha)\sin(\beta)\right)\\ & =\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \end{aligned} $$

Proof. $$\begin{aligned} \sin(\alpha+\beta) & =\Im\left(e^{i(\alpha+\beta)}\right) & i & =\sqrt{-1}\\ & =\Im\left(e^{i\alpha}e^{i\beta}\right)\\ & =\Im\left((\cos(\alpha)+i\sin(\alpha)(\cos(\beta)+i\sin(\beta)\right) & i^{2} & =-1\\ & =\Im\left(\cos(\alpha)\cos(\beta)+i\left(\cos(\alpha)\sin(\beta)+\sin(\alpha)\cos(\beta)\right)+i^{2}\sin(\alpha)\sin(\beta)\right)\\ & =\cos(\alpha)\sin(\beta)+\sin(\alpha)\cos(\beta) \end{aligned} $$

Hyperbolic Functions in Exponential Form

$$ \begin{aligned} \cosh(x) & :=\dfrac{e^{x}+e^{-x}}{2} & \text{sech}(x) & =\dfrac{1}{\cosh(x)}\\ \sinh(x) & :=\dfrac{e^{x}-e^{-x}}{2} & \text{csch}(x) & =\dfrac{1}{\sinh(x)}\\ \tanh(x) & :=\dfrac{\sinh(x)}{\cosh(x)} & \coth(x) & =\dfrac{1}{\tanh(x)} \end{aligned} $$

More Necessary Identities

$$ \begin{aligned} \cosh^{2}(x)-\sinh^{2}(x) & =1 & \cosh^{2}(x) & =\dfrac{\cosh(2x)+1}{2}\\ \sinh(2x) & =2\sinh(x)\cosh(x) & \sinh^{2}(x) & =\dfrac{\cosh(2x)-1}{2}\\ \cosh(2x) & =\cosh^{2}(x)+\sinh^{2}(x) & \tanh^{2}(x) & =1-\text{sech}^{2}(x) \end{aligned} $$

Derivative of the Hyperbolic Functions

$$ \begin{aligned} \frac{d}{dx}\left[\sinh(x)\right] & =\cosh(x) & \frac{d}{dx}\left[\cosh(x)\right] & =\sinh(x)\\ \frac{d}{dx}\left[\tan(x)\right] & =\text{sech}^{2}(x) & \frac{d}{dx}\left[\coth(x)\right] & =-\text{csch}^{2}(x)\\ \frac{d}{dx}\left[\text{sech}(x)\right] & =-\text{sech}(x)\tanh(x) & \frac{d}{dx}\left[\text{csch}(x)\right] & =-\text{csch}(x)\coth(x) \end{aligned} $$

Integration Table for Hyperbolic Functions

$$ \begin{aligned} \int\sinh(x)\,dx & =\cosh(x)+C & \int\cosh(x)\,dx & =\sinh(x)+C\\ \int\text{sech}^{2}(x)\,dx & =\tan(x)+C & \int-\text{csch}^{2}(x)\,dx & =\coth(x)+C\\ \int-\text{sech}(x)\tanh(x)\,dx & =\text{sech}(x)+C & \int-\text{csch}(x)\coth(x)\,dx & =\text{csch}(x)+C \end{aligned} $$

Inverse Hyperbolic Functions

  1. Let $x\in(-\infty,\infty)$ and $f(x)=\sinh(x)\in(-\infty,\infty)$, then $f^{-1}(x)=\sinh^{-1}(x)\in(-\infty,\infty)$ where $x\in(-\infty,\infty)$.
  2. Let $x\in[0,\infty)$ and $f(x)=\cosh(x)\in[1,\infty)$, then $f^{-1}(x)=\cosh^{-1}(x)\in[0,\infty)$ where $x\in[1,\infty)$.
  3. Let $x\in(0,1]$ and $f(x)=\text{sech}(x)\in[0,\infty)$, then $f^{-1}(x)=\text{sech}^{-1}(x)\in(0,1]$ where $x\in[0,\infty)$.
  4. Let $x\in(-\infty,\infty)$ and $f(x)=\tanh(x)\in(-1,1)$, then $f^{-1}(x)=\tanh^{-1}(x)\in(-\infty,\infty)$ where $x\in(-1,1)$.
  5. Let $x\in(-\infty,0)\cup(0,\infty)$ and $f(x)=\coth(x)\in(-\infty,-1)\cup(1,\infty)$, then $f^{-1}(x)=\coth^{-1}(x)$ where $x\in(-\infty,-1)\cup(1,\infty)$.
  6. Let $x\in(-\infty,0)\cup(0,\infty)$ and $f(x)=\text{csch}^{-1}(x)\in(-\infty,0)\cup(0,\infty)$, then $f^{-1}(x)=\text{csch}^{-1}(x)$ where $x\in(-\infty,0)\cup(0,\infty)$. $$ \begin{aligned} \text{sech}^{-1}(x) & =\cosh^{-1}(\frac{1}{x})\\ \text{csch}^{-1}(x) & =\sinh^{-1}(\frac{1}{x})\\ \coth^{-1}(x) & =\tanh^{-1}(\frac{1}{x}) \end{aligned} $$

Derivative of the Inverse Hyperbolic Functions

$$ \begin{aligned} \frac{d}{dx}\left[\sinh^{-1}(x)\right] & =\dfrac{1}{\sqrt{1+x^{2}}}\\ \ddx{\cosh^{-1}(x)} & =\dfrac{1}{\sqrt{x^{2}-1}} & x & >1\\ \ddx{\text{sech}^{-1}(x)} & =-\dfrac{1}{x\sqrt{1-x^{2}}} & |x| & <1\\ \ddx{\tanh^{-1}(x)} & =\dfrac{1}{1-x^{2}} & |x| & >1\\ \ddx{\coth^{-1}(x)} & =\dfrac{1}{1-x^{2}} & 0<x & <1\\ \ddx{\text{csch}^{-1}(x)} & =-\dfrac{1}{|x|\sqrt{1+x^{2}}} & x & \ne0 \end{aligned} $$

Integral Table of Invese Hyperbolic Functions

$$ \begin{aligned} \int\dfrac{1}{\sqrt{1+x^{2}}}\,dx & =\sinh^{-1}(x)+C\\ \int\dfrac{1}{\sqrt{x^{2}-1}}\,dx & =\cosh^{-1}(x)+C & x & >0\\ \int-\dfrac{1}{x\sqrt{1-x^{2}}}\,dx & =\text{sech}^{-1}(x)+C\\ \int\dfrac{1}{1-x^{2}}\,dx & =\tanh^{-1}(x)+C\\ \int\dfrac{1}{1-x^{2}}\,dx & =\coth^{-1}(x)+C & x & >0\\ \int-\dfrac{1}{|x|\sqrt{1+x^{2}}}\,dx & =\text{csch}^{-1}(x)+C & x & \ne0 \end{aligned} $$

Examples

Evaluate an integral

$$\int\dfrac{2\,dx}{\sqrt{3+4x^{2}}}$$$$ \begin{aligned} \int\dfrac{2\,dx}{\sqrt{3+4x^{2}}} & =\int\dfrac{2\,dx}{\sqrt{(\sqrt{3})^{2}+(2x)^{2}}}\\ & =\int\dfrac{2\,dx}{\sqrt{(\sqrt{3})^{2}+\left(\dfrac{\sqrt{3}}{\sqrt{3}}\right)^{2}(2x)^{2}}}\\ & =\int\dfrac{2\,dx}{\sqrt{(\sqrt{3})^{2}\left(1+\left(\dfrac{2x}{\sqrt{3}}\right)^{2}\right)}}\\ & =\dfrac{2}{\sqrt{3}}\int\dfrac{dx}{\sqrt{1+\left(\dfrac{2x}{\sqrt{3}}\right)^{2}}}\\ u & =\dfrac{2}{\sqrt{3}}x\\ du & =\frac{2}{\sqrt{3}}\,dx\\ \frac{\sqrt{3}}{2}du & =dx\\ & =\frac{2}{\sqrt{3}}\cdot\frac{\sqrt{3}}{2}\int\dfrac{du}{\sqrt{1+u^{2}}}\\ & =\sinh^{-1}(u)+C\\ & =\sinh^{-1}(\frac{2x}{\sqrt{3}})+C \end{aligned} $$
In [3]:
integrate(2/sqrt(3+4*x^2),x);
Out[3]:
\[\tag{${\it \%o}_{3}$}{\rm asinh}\; \left({{2\,x}\over{\sqrt{3}}}\right)\]
In [ ]: