sech function
- Categories:
- hyperbolic functions
- pure mathematics
The sech function is a hyperbolic function. It is also known as the hyperbolic secant function.
Here is a video that explains the cosech, sech, and coth functions:
Equation and graph
The sech function is the reciprocal of the cosh function:
$$ \operatorname{sech}{x} = \frac{1}{cosh{x}} $$
(This is analogous with trig functions, where sec is the reciprocal of cos.)
Substituting the formula for cosh gives:
$$ \operatorname{sech}{x} = \frac{2}{e^{x}+e^{-x}} $$
Here is a graph of the function:
Sketching sech
We can sketch the sech curve based on the cosh curve:
The cosh curve tends to +∞ for large negative $x$, it has a value of 1 when $x = 0$, and it tends to +∞ for large positive $x$.
The sech curve is the reciprocal of cosh. It tends to zero for large negative $x$, it has a value of 1 when $x = 0$, and it tends to +∞ for large positive $x$.
Other forms of the equation
If we take the two alternative forms of the cosh function, and invert them, we get two alternative forms of the sech function:
$$ \operatorname{sech}{x} = \frac{2e^{x}}{e^{2x}+1} $$
$$ \operatorname{sech}{x} = \frac{2e^{-x}}{1+e^{-2x}} $$