gives the inverse hyperbolic tangent of the complex number .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, ArcTanh automatically evaluates to exact values.
- ArcTanh can be evaluated to arbitrary numerical precision.
- ArcTanh automatically threads over lists.
- ArcTanh[z] has branch cut discontinuities in the complex plane running from to and to .
- ArcTanh can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcTanh is the inverse hyperbolic tangent function. For a real number x, ArcTanh[x] represents the hyperbolic angle measure such that .
- ArcTanh automatically threads over lists. For certain special arguments, ArcTanh automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcTanh may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcTanh include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcTanh is defined for complex argument by . ArcTanh[z] has branch cut discontinuities in the complex plane.
- Related mathematical functions include Tanh, ArcCoth, and ArcTan.
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion about 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 ArcTanh efficiently at high precision:
ArcTanh threads elementwise over lists and matrices:
ArcTanh can be used with Interval and CenteredInterval objects:
Specific Values (4)
Plot the ArcTanh function:
Function Properties (11)
ArcTanh is defined for all real values from the interval :
ArcTanh achieves all real values:
Function range for arguments from the complex domain:
ArcTanh is an odd function:
ArcTanh is not an analytic function:
ArcTanh is increasing over its real domain:
ArcTanh is injective:
ArcTanh is surjective:
ArcTanh is neither non-negative nor non-positive:
It has both singularity and discontinuity in (-∞,-1] and [1,∞):
ArcTanh is neither convex nor concave:
Series Expansions (4)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCoth:
Representation through inverse Jacobi functions:
ArcTanh is a special case of the hypergeometric function Hypergeometric2F1:
ArcTanh can be represented in terms of MeijerG:
ArcTanh can be represented as a DifferentialRoot:
Find the rapidity corresponding to a speed of 0.999 times the speed of light:
Branch cuts of ArcTanh:
Wolfram Research (1988), ArcTanh, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTanh.html (updated 2021).
Wolfram Language. 1988. "ArcTanh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcTanh.html.
Wolfram Language. (1988). ArcTanh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTanh.html