# ArcSech ArcSech[z]

gives the inverse hyperbolic secant of the complex number .

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

# Background & Context

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

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(40)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate ArcSech efficiently at high precision:

ArcSech threads elementwise over lists and matrices:

ArcSech can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Values of ArcSech at fixed points:

Values at infinity:

Zero of ArcSech:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcSech function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(10)

ArcSech is defined for all real values from the interval :

Complex domain:

ArcSech achieves all real values greater than or equal to 0:

Function range for arguments from the complex domain:

ArcSech is not an analytic function:

Nor is it meromorphic:

ArcSech is decreasing over its real domain:

ArcSech is injective:

ArcSech is not surjective:

ArcSech is non-negative over its real domain:

It has both singularity and discontinuity in (-,0] and [1,):

ArcSech is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcSech:

Definite integral of ArcSech over its real domain:

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcSech around :

Find series expansions at branch points and branch cuts:

Apply ArcSech to a power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcSech:

Use TrigToExp to express in terms of logarithm:

Convert back:

Expand assuming real variables and :

### Function Representations(5)

Represent using ArcCosh:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

ArcSech can be represented in terms of MeijerG:

ArcSech can be represented as a DifferentialRoot:

## Applications(2)

Branch cut line runs along the real axis:

Solve a differential equation:

## Properties & Relations(2)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the ArcSech: