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 .
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:
Asymptotic expansion at a singular point:
Numerical Evaluation (6)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate ArcTanh efficiently at high precision:
ArcTanh can deal with real‐valued intervals:
ArcTanh threads elementwise over lists and matrices:
Specific Values (4)
Values of ArcTanh at fixed points:
Values at infinity:
Zero of ArcTanh:
Find the value of satisfying equation :
Substitute in the value:
Visualize the result:
Plot the ArcTanh function:
Plot the real part of :
Plot the imaginary part of :
Polar plot with :
Function Properties (4)
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:
Formula for the derivative:
Indefinite integral of ArcTanh:
Definite integral of ArcTanh over an interval centered at the origin is 0:
Series Expansions (4)
Find the Taylor expansion using Series:
Plots of the first three approximations for ArcTanh around :
General term in the series expansion of ArcTanh:
Find series expansions at branch points and branch cuts:
ArcTanh can be applied to power series:
Function Identities and Simplifications (3)
Simplify expressions involving ArcTanh:
Express ArcTanh using Log:
Expand assuming real variables and :
Function Representations (5)
Find the rapidity corresponding to a speed of 0.999 times the speed of light:
Branch cuts of ArcTanh:
Properties & Relations (2)
Express ArcTanh using Log:
ArcTanh is a special case of some special functions: