# ArcCsc ArcCsc[z]

gives the arc cosecant of the complex number .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• All results are given in radians.
• For real z outside the interval to , the results are always in the range to , excluding 0.
• For certain special arguments, ArcCsc automatically evaluates to exact values.
• ArcCsc can be evaluated to arbitrary numerical precision.
• ArcCsc automatically threads over lists.
• ArcCsc[z] has a branch cut discontinuity in the complex plane running from to .
• ArcCsc can be used with Interval and CenteredInterval objects. »

# Background & Context

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

# 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 ArcCsc efficiently at high precision:

ArcCsc threads elementwise over lists and matrices:

ArcCsc can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Values of ArcCsc at fixed points:

Values at infinity:

Singular points of ArcCsc:

ArcCsc is not differentiable at these points:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

### Visualization(3)

Plot the ArcCsc function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### Function Properties(11)

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

Complex domain:

ArcCsc achieves all real values from the interval except 0:

Function range for arguments from the complex domain:

ArcCsc is an odd function:

ArcCsc is not an analytic function:

Nor is it meromorphic:

ArcCsc is monotonic in a specific range:

ArcCsc is injective:

ArcCsc is not surjective:

ArcCsc is neither non-negative nor non-positive:

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

ArcCsc is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of ArcCsc:

Definite integral of ArcCsc over the interval :

More integrals:

### Series Expansions(3)

Find the Taylor expansion using Series:

Plots of the first three approximations for ArcCsc around :

Find series expansions at branch points and branch cuts:

ArcCsc can be applied to power series:

### Function Identities and Simplifications(3)

Simplify expressions involving ArcCsc:

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the ArcCsc:

Use TrigToExp to express in terms of logarithm:

Use ExpToTrig to convert back:

### Function Representations(5)

Represent using ArcSin:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

ArcCsc can be represented in terms of MeijerG:

ArcCsc can be represented as a DifferentialRoot:

## Applications(3)

Branch cut of ArcCsc runs along the real axis:

Solve a differential equation:

Distribution of the average distance s of all pairs of points in a ddimensional hypersphere:

Lowdimensional distributions can be expressed in elementary functions:

Plot these distributions:

## Properties & Relations(4)

Compose with inverse functions:

Use PowerExpand to disregard multivaluedness of the ArcCsc: