BUILT-IN MATHEMATICA SYMBOL

ArcSinh

ArcSinh[z]
gives the inverse hyperbolic sine

of the complex number

.

MORE INFORMATION

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 .

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Series expansion:

Evaluate numerically:

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Out[1]=

In[1]:=

Out[1]=

Series expansion:

In[1]:=

Out[1]=

Scope (6)

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

ArcSinh threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformation is automatically applied:

TraditionalForm formatting:

Generalizations & Extensions (3)

ArcSinh can be applied to a power series:

ArcSinh can deal with real-valued intervals:

Infinite arguments generate exact results:

Applications (2)

Compute the length of hyperbola

from the base to given

:

Solve a differential equation:

Properties & Relations (5)

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the ArcSinh:

Alternatively, evaluate under additional assumptions:

Use TrigToExp to express ArcSinh using logarithm:

Use Reduce to solve an equation in terms of ArcSinh:

ArcSinh is a special case of some special functions:

Integrals:

Possible Issues (2)

Generically

:

When using input in traditional form, parentheses are needed around the argument:

Neat Examples (1)

Compute 100,000 digits of

, and show the first and last 20 digits: