Martin McBride, 2021-02-04

Tags sech cosh

Categories hyperbolic functions pure mathematics

The sech function is a hyperbolic function. It is also known as the *hyperbolic secant* function.

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:

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 $x$.

If we take the two alternative forms of the cosh function, and invert them, we get two alternative forms of the cosech function. First:

$$ \operatorname{cosech}{x} = \frac{2e^{x}}{e^{2x}+1} $$

Alternatively, if we multiply the top and bottom of the original equation for the sinh function by $e^{-x}$ we get:

$$ \operatorname{cosech}{x} = \frac{2e^{-x}}{1+e^{-2x}} $$

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