# Documentation

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# acsch

Inverse hyperbolic cosecant

Y = acsch(X)

## Description

Y = acsch(X) returns the inverse hyperbolic cosecant for each element of X.

The acsch 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 cosecant function over the domains and .

x1 = -20:0.01:-1;
x2 = 1:0.01:20;
plot(x1,acsch(x1),x2,acsch(x2))
grid on
xlabel('x')
ylabel('y')

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### Inverse Hyperbolic Cosecant

For real values $x$ in the domain $x<0$ and $x>0$, the inverse hyperbolic cosecant satisfies

${\text{csch}}^{-1}\left(z\right)={\mathrm{sinh}}^{-1}\left(\frac{1}{z}\right)=\mathrm{log}\left(\frac{1}{x}+\sqrt{\frac{1}{{x}^{2}}+1}\right).$

For complex numbers $z=x+iy$, the call acsch(z) returns complex results.