# ArcSec ArcSec[z]

gives the arc secant of the complex number .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• All results are given in radians.
• For real outside the interval to , the results are always in the range to , excluding .
• For certain special arguments, ArcSec automatically evaluates to exact values.
• ArcSec can be evaluated to arbitrary numerical precision.
• ArcSec automatically threads over lists.
• ArcSec[z] has a branch cut discontinuity in the complex plane running from to .
• ArcSec can be used with Interval and CenteredInterval objects. »

# Background & Context

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

# Examples

open allclose all

## Basic Examples(5)

Divide by Degree to get results in degrees:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(41)

### 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 ArcSec efficiently at high precision:

ArcSec threads elementwise over lists and matrices:

ArcSec can be used with Interval and CenteredInterval objects:

### Specific Values(5)

Values of ArcSec at fixed points:

Simple exact values are generated automatically:

Values at infinity:

Zero of ArcSec:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcSec function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(10)

ArcSec is defined for all real values except from the interval :

Complex domain:

ArcSec lies between and :

Function range for arguments from the complex domain:

ArcSec is not an analytic function:

Nor is it meromorphic:

ArcSec is monotonic in a specific range:

ArcSec is injective:

ArcSec is not surjective:

ArcSec is non-negative on its real domain:

It has both singularity and discontinuity for x in [-1,1]:

ArcSec is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcSec:

Definite integral over the interval :

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcSec around :

Find series expansions at branch points and branch cuts:

ArcSec can be applied to power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcSec:

Use TrigToExp to express in terms of logarithm:

Use ExpToTrig to convert back:

Expand assuming real variables and :

### Function Representations(5)

Represent using ArcCos:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

ArcSec can be represented in terms of MeijerG:

ArcSec can be represented as a DifferentialRoot:

## Applications(2)

Branch cut of ArcSec runs along the real axis:

Solve a differential equation:

## Properties & Relations(4)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the ArcSec: