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)
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
Evaluate ArcTanh efficiently at high precision:
ArcTanh threads elementwise over lists and matrices:
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:
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:
ArcTanh is neither convex nor concave:
Series Expansions (4)
Function Identities and Simplifications (3)
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