# ArcSinh

ArcSinh[z]

gives the inverse hyperbolic sine of the complex number .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, ArcSinh automatically evaluates to exact values.
• ArcSinh can be evaluated to arbitrary numerical precision.
• ArcSinh automatically threads over lists.
• ArcSinh[z] has branch cut discontinuities in the complex plane running from to and to .

# Background & Context

• ArcSinh is the inverse hyperbolic sine function. For a real number , ArcSinh[x] represents the hyperbolic angle measure such that .
• ArcSinh automatically threads over lists. For certain special arguments, ArcSinh automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcSinh may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcSinh include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
• ArcSinh is defined for complex argument by . ArcSinh[z] has branch cut discontinuities in the complex plane.
• Related mathematical functions include Sinh, ArcCosh, and ArcSin.

# Examples

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## Basic Examples(3)

Evaluate numerically:

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Plot over a subset of the reals:

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Series expansion:

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