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{title2=$\operatorname{arcsinh}z$}

The solutions of $\sinh w=z$ are
$$
w=\operatorname{Arcsinh}z+2\pi ik
\quad\hbox{or}\quad
w=-\operatorname{Arcsinh}z+(2k+1)\pi i,
\qquad k\in\mathbb Z,
$$
for any chosen local branch $\operatorname{Arcsinh}$.

It is the complex multivalued sine member of the <inverse hyperbolic functions>.

Multivalued inverse hyperbolic sine

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 7E / Solution

Multivalued inverse hyperbolic sine ( $arcsinh z$ )

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Multivalued inverse hyperbolic sine (arcsinhz)

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Multivalued inverse hyperbolic sine ( $arcsinh z$ )

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