ArcCoth
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcCoth automatically evaluates to exact values.
- ArcCoth can be evaluated to arbitrary numerical precision.
- ArcCoth automatically threads over lists.
- ArcCoth[z] has a branch cut discontinuity in the complex
plane running from
to
.
- ArcCoth can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcCoth is the inverse hyperbolic cotangent function. For a real number
, ArcCoth[x] represents the hyperbolic angle measure
such that
.
- ArcCoth automatically threads over lists. For certain special arguments, ArcCoth automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCoth may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCoth include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcCoth is defined for complex argument
by
. ArcCoth[z] has a branch cut discontinuity in the complex
plane.
- Related mathematical functions include ArcTanh, Coth, and ArcCot.
Examples
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0tzy57lg-h4qo34

Plot over a subset of the reals:

https://wolfram.com/xid/0tzy57lg-bh7r47

Plot over a subset of the complexes:

https://wolfram.com/xid/0tzy57lg-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0tzy57lg-fdvdfy

Asymptotic expansion at Infinity:

https://wolfram.com/xid/0tzy57lg-uoije8

Asymptotic expansion at a singular point:

https://wolfram.com/xid/0tzy57lg-bzvcs4

Scope (43)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0tzy57lg-pbj


https://wolfram.com/xid/0tzy57lg-gz8fud

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

https://wolfram.com/xid/0tzy57lg-gvz0ta

Evaluate for complex arguments:

https://wolfram.com/xid/0tzy57lg-b7vho4

Evaluate ArcCoth efficiently at high precision:

https://wolfram.com/xid/0tzy57lg-di5gcr


https://wolfram.com/xid/0tzy57lg-bq2c6r

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

https://wolfram.com/xid/0tzy57lg-e3cimb


https://wolfram.com/xid/0tzy57lg-gnclzo


https://wolfram.com/xid/0tzy57lg-lmyeh7

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0tzy57lg-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0tzy57lg-thgd2

Or compute the matrix ArcCoth function using MatrixFunction:

https://wolfram.com/xid/0tzy57lg-o5jpo

Specific Values (5)
Values of ArcCoth at fixed points:

https://wolfram.com/xid/0tzy57lg-nww7l


https://wolfram.com/xid/0tzy57lg-gp8fq9


https://wolfram.com/xid/0tzy57lg-drqkdo

Singular points of ArcCoth:

https://wolfram.com/xid/0tzy57lg-cw39qs


https://wolfram.com/xid/0tzy57lg-2xdvwq

Find the value of satisfying equation
:

https://wolfram.com/xid/0tzy57lg-f2hrld


https://wolfram.com/xid/0tzy57lg-gla6mn

Simple exact values are generated automatically:

https://wolfram.com/xid/0tzy57lg-dxuy0g


https://wolfram.com/xid/0tzy57lg-gy9gt3

Visualization (3)
Plot the ArcCoth function:

https://wolfram.com/xid/0tzy57lg-ecj8m7


https://wolfram.com/xid/0tzy57lg-bo5grg


https://wolfram.com/xid/0tzy57lg-b75uqy


https://wolfram.com/xid/0tzy57lg-epb4bn

Function Properties (11)
ArcCoth is defined for all real values except from the interval :

https://wolfram.com/xid/0tzy57lg-cl7ele


https://wolfram.com/xid/0tzy57lg-de3irc

ArcCoth achieves all real values except 0:

https://wolfram.com/xid/0tzy57lg-evf2yr

Function range for arguments from the complex domain:

https://wolfram.com/xid/0tzy57lg-fphbrc

ArcCoth is an odd function:

https://wolfram.com/xid/0tzy57lg-h9i4y4

ArcCoth is not an analytic function:

https://wolfram.com/xid/0tzy57lg-h5x4l2


https://wolfram.com/xid/0tzy57lg-e434t9

ArcCoth is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0tzy57lg-g6kynf

ArcCoth is injective:

https://wolfram.com/xid/0tzy57lg-gi38d7


https://wolfram.com/xid/0tzy57lg-ctca0g

ArcCoth is not surjective:

https://wolfram.com/xid/0tzy57lg-hkqec4


https://wolfram.com/xid/0tzy57lg-b1r9xi

ArcCoth is neither non-negative nor non-positive:

https://wolfram.com/xid/0tzy57lg-84dui

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

https://wolfram.com/xid/0tzy57lg-mdtl3h


https://wolfram.com/xid/0tzy57lg-mn5jws

ArcCoth is neither convex nor concave:

https://wolfram.com/xid/0tzy57lg-kdss3

TraditionalForm formatting:

https://wolfram.com/xid/0tzy57lg-k8e7of

Differentiation (3)

https://wolfram.com/xid/0tzy57lg-mmas49


https://wolfram.com/xid/0tzy57lg-nfbe0l


https://wolfram.com/xid/0tzy57lg-fxwmfc


https://wolfram.com/xid/0tzy57lg-odmgl1

Integration (3)
Indefinite integral of ArcCoth:

https://wolfram.com/xid/0tzy57lg-bponid

A definite integral involving ArcCoth:

https://wolfram.com/xid/0tzy57lg-b9jw7l


https://wolfram.com/xid/0tzy57lg-rkknd0


https://wolfram.com/xid/0tzy57lg-bk07nd

Series Expansions (4)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0tzy57lg-ewr1h8

Plot the first three approximations for ArcCoth around :

https://wolfram.com/xid/0tzy57lg-binhar

General term in the series expansion of ArcCoth:

https://wolfram.com/xid/0tzy57lg-z1i2ky

Find series expansions at branch points and branch cuts:

https://wolfram.com/xid/0tzy57lg-ybj


https://wolfram.com/xid/0tzy57lg-ceo3xp

ArcCoth can be applied to power series:

https://wolfram.com/xid/0tzy57lg-i5r8gq

Function Identities and Simplifications (3)
Simplify expressions involving ArcCoth:

https://wolfram.com/xid/0tzy57lg-gji1fc


https://wolfram.com/xid/0tzy57lg-dut2dg


https://wolfram.com/xid/0tzy57lg-ll1xgl

Expand assuming real variables and
:

https://wolfram.com/xid/0tzy57lg-se6

Function Representations (5)
Represent using ArcCoth:

https://wolfram.com/xid/0tzy57lg-osnk6u

Representation through inverse Jacobi functions:

https://wolfram.com/xid/0tzy57lg-m9whnz


https://wolfram.com/xid/0tzy57lg-i0p6v1

ArcCoth is a special case of Hypergeometric2F1:

https://wolfram.com/xid/0tzy57lg-ds7bqo

ArcCoth can be represented in terms of MeijerG:

https://wolfram.com/xid/0tzy57lg-ogbpj


https://wolfram.com/xid/0tzy57lg-fhqwts

ArcCoth can be represented as a DifferentialRoot:

https://wolfram.com/xid/0tzy57lg-zbzsc

Applications (3)Sample problems that can be solved with this function
Branch cuts of ArcCoth:

https://wolfram.com/xid/0tzy57lg-b2lm4s

Solve a differential equation:

https://wolfram.com/xid/0tzy57lg-cdxipn

Compute the probability that a random variable is within one standard deviation from the mean:

https://wolfram.com/xid/0tzy57lg-mwroh2

https://wolfram.com/xid/0tzy57lg-cyxo85

https://wolfram.com/xid/0tzy57lg-b1pn7q

Get the numeric value of this probability:

https://wolfram.com/xid/0tzy57lg-bcm1f2

Plot the PDF with this region:

https://wolfram.com/xid/0tzy57lg-k562of

Wolfram Research (1988), ArcCoth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCoth.html (updated 2021).
Text
Wolfram Research (1988), ArcCoth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCoth.html (updated 2021).
Wolfram Research (1988), ArcCoth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCoth.html (updated 2021).
CMS
Wolfram Language. 1988. "ArcCoth." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCoth.html.
Wolfram Language. 1988. "ArcCoth." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcCoth.html.
APA
Wolfram Language. (1988). ArcCoth. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCoth.html
Wolfram Language. (1988). ArcCoth. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCoth.html
BibTeX
@misc{reference.wolfram_2025_arccoth, author="Wolfram Research", title="{ArcCoth}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCoth.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_arccoth, organization={Wolfram Research}, title={ArcCoth}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCoth.html}, note=[Accessed: 29-March-2025
]}