gives the inverse hyperbolic cotangent of the complex number .
- 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.
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcCoth efficiently at high precision:
ArcCoth threads elementwise over lists and matrices:
ArcCoth can be used with Interval and CenteredInterval objects:
Specific Values (5)
Plot the ArcCoth function:
Function Properties (11)
ArcCoth is defined for all real values except from the interval :
ArcCoth achieves all real values except 0:
Function range for arguments from the complex domain:
ArcCoth is an odd function:
ArcCoth is not an analytic function:
ArcCoth is neither non-decreasing nor non-increasing:
ArcCoth is injective:
ArcCoth is not surjective:
ArcCoth is neither non-negative nor non-positive:
It has both singularity and discontinuity in [-1,1]:
ArcCoth is neither convex nor concave:
Series Expansions (4)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCoth:
Representation through inverse Jacobi functions:
ArcCoth is a special case of Hypergeometric2F1:
ArcCoth can be represented in terms of MeijerG:
ArcCoth can be represented as a DifferentialRoot:
Branch cuts of ArcCoth:
Solve a differential equation:
Compute the probability that a random variable is within one standard deviation from the mean:
Get the numeric value of this probability:
Wolfram Research (1988), ArcCoth, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCoth.html (updated 2021).
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. Retrieved from https://reference.wolfram.com/language/ref/ArcCoth.html