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Hyperbolic secant function

The ratio of number $2$ to sum of the negative and positive natural exponential functions is called the hyperbolic secant function.


$e$ is a positive irrational mathematical constant and $x$ is a variable. The positive and negative natural exponential functions are expressed as $e^x$ and $e^{-x}$ respectively in mathematics.

The sum of the positive and negative natural exponential functions is equal to $e^x+e^{-x}$

The ratio of the quantity $2$ to the summation of them is written mathematically as follows.

$\large \dfrac{2}{e^x+e^{-x}}$

Mathematically, the ratio is called the hyperbolic secant function. The hyperbolic secant is represented by ${\mathop{\rm sech}\nolimits}$ but the function is in terms of $x$. Hence, the hyperbolic secant function is represented by ${\mathop{\rm sech}\nolimits}{x}$ in mathematics.

$\large {\mathop{\rm sech}\nolimits}{x} \,=\, \dfrac{2}{e^x+e^{-x}}$

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