W3 Hyperbolic Functions

Even/Odd Functions

• Even Function
  • $\boxed{f(-x)=f(x)}$
  • symmetrical over $y$-axis
• Odd Function
  • $\boxed{f(-x)=-f(x)}$
  • flipped over $y$ and $x$ axis

Hyperbolic Functions

graph	function	domain($x$)	range($y$)
	$\sinh x=\frac{e^x-e^{-x}}{2}$	$\R$	$\R$
	$\cosh x=\frac{e^x+e^{-x}}{2}$	$\R$	$[1,\infty]$
	$\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$	$\R$	$(-1,1)$

Reciprocal Hyperbolic Functions

graph	function	domain($x$)	range($y$)
	$\textmd{csch }x=\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}$	$\R$ \ $\{0\}$	$\R$ \ $\{0\}$
	$\textmd{sech }x=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$	$\R$	$( 0, 1] \,$
	$\coth x=\frac{\cosh x}{\sinh x}$	$(-\infty,0)\cup(0, \infty)$	$(-\infty,-1)\cup(1), \infty)$

Inverse Hyperbolic Functions

Derivation: Derivation of Inverse Hyperbolic Functions

• $\textmd{arcsinh }u=\log (u+\sqrt{u^2+1})\frac{du}{dx}$

• $\textmd{arccosh }u=\log (u+\sqrt{u^2-1})\frac{du}{dx}$

• $\textmd{arctanh }u=\frac{1}{3}\log(\frac{1+u}{1-u})\frac{du}{dx}$

• $\textmd{arccsch }u=-\frac{1}{|u|\sqrt{1+u^2}}\frac{du}{dx}$

• $\textmd{arcsech }u=-\frac{1}{u\sqrt{1-u^2}}\frac{du}{dx}$

• $\textmd{arccoth }u=\frac{1}{1-u^2}\frac{du}{dx}$

Inverse Reciprocal Hyperbolic Functions

Hyperbolic Formulae

• $\tanh x=\frac{\sinh x}{\cosh x}$

Sum/Difference Formulae

• $\sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y$
• $\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y$
• $\sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y$
• $\cosh(x-y)=\cosh x\cosh y=\sinh x\sinh y$

Double angle Formulae

• $\sinh(2x)=2\sinh(x)\cosh(x)$
• $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$
• $\cosh(2x)=2\cosh^2(x)-1$
• $\cosh(2x)=2\sinh^2(x)+1$

Basic Identities

• $\cosh^2(x)-\sinh^2(x)=1$
• $\coth^2(x)-1=\textmd{cosech}^2(x)$
• $1-\tanh^2(x)=\textmd{sech}^2(x)$

Derivatives

• $\frac{d}{dx}(\sinh u)=\cosh u \times\frac{du}{dx}$
• $\frac{d}{dx}(\textmd{cosh }u)=\sinh u \times\frac{du}{dx}$
• $\frac{d}{dx}(\tanh u)=\textmd{sech}^2u \times\frac{du}{dx}$

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• $\frac{d}{dx}(\textmd{csch }u)=-\textmd{csch }u\times\frac{du}{dx}$
• $\frac{d}{dx}(\textmd{sech }u)=-\textmd{sech }u\tanh u\frac{du}{dx}$
• $\frac{d}{dx}(\textmd{coth }u)=-\textmd{csch}^2u\frac{du}{dx}$

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• $\frac{d}{dx}(\textmd{arcsinh }x)=\frac{1}{\sqrt{x^2+1}}$
• $\frac{d}{dx}(\textmd{arccosh }x)=\frac{1}{\sqrt{x^2-1}}$
• $\frac{d}{dx}(\textmd{arctanh }x)=\frac{1}{1-x^2}$