# Documentation

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# asech

Inverse hyperbolic secant

## Syntax

```Y = asech(X) ```

## Description

`Y = asech(X)` returns the inverse hyperbolic secant for each element of `X`.

The `asech` function operates element-wise on arrays. The function's domains and ranges include complex values. All angles are in radians.

## Examples

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Graph the inverse hyperbolic secant over the domain .

```x = 0.01:0.001:1; plot(x,asech(x)) grid on xlabel('x') ylabel('y')```

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### Inverse Hyperbolic Secant

For real values $x$ in the domain $0, the inverse hyperbolic secant satisfies

`${\text{sech}}^{-1}\left(x\right)={\mathrm{cosh}}^{-1}\left(\frac{1}{x}\right)=\mathrm{log}\left(\frac{1}{x}+\sqrt{\frac{1}{{x}^{2}}-1}\right).$`

For complex numbers $z=x+iy$ as well as real values in the regions $-\text{\hspace{0.17em}}\infty and $1\le z<\infty$, the call `asech(z)` returns complex results.