Hyperbolic Function

The hyperbolic functions are a family of functions that are named after the hyperbola shape that occurs when they are parameterised with $t$. They aid in Derivation & Integration of certain functions, as well as connect to Complex Number.

Definitions §

$\sinh (x)$ §

An odd function, where f(-x) = -f(x), the hyperbolic sine function is defined as:

$\sinh (x)=\frac{1}{2}(e^x-e^{-x})$

• $\sinh (0)=0,\forall x\in \mathbb{R}$
• $\sinh (-x)=-\sinh (x)$

$\cosh (x)$ §

An even function, the hyperbolic cosine function is defined as:

$\cosh (x)=\frac{1}{2}(e^x+e^{-x})$

• $\cosh (0)=1$
• $\cosh (-x)=\cosh (x)$

$\tanh (x)$ §

Just like the trigonometric equivalent, the hyperbolic tangent is defined as the quotient between the sinh and cosh:

$\tanh (x)=\frac{\sinh (x)}{\cosh (x)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$

• $\tanh (0)=0$
• $\tanh (-x)=-\tanh (x)$

Why are it called hyperbolic functions? §

The rule for a hyperbola is $y^2-x^2=1$ (for horizontal hyperbola) or $x^2-y^2=1$ (for vertical hyperbola).

Let $x=\cosh (t)$ and $y=\sinh (t)$. Then:

$x^2-y^2=\cosh (t{)}^2-\sinh (t{)}^2$ $=(\frac{1}{2}(e^x+e^{-x}){)}^2-(\frac{1}{2}(e^x-e^{-x}){)}^2$ $=\frac{1}{4}[(e^x+e^{-x}{)}^2-(e^{2x}-e^{-2x}{)}^2]$ $=\frac{1}{4}[e^{2x}+2e^0+e^{-2x}-(e^{2x}-2e^0+e^{-2x})]$ $\frac{1}{4}[e^{2x}-e^{2x}+e^{-2x}-e^{-2x}+2e^0+2e^0]$ $\frac{1}{4}[4e^0]=\frac{1}{4}\times 4=1$

$\therefore \cosh (t{)}^2-\sinh (t{)}^2=1=x^2-y^2$

Gallery of Hyperbolic Functions §

y = \cosh{x}
y = \sinh{x}
y = \tanh{x}

Green: $\sinh x$ Blue: $\cosh x$ Purple: $\tanh x$