Inverse Hyperbolic Tangent

Online calculator for calculating the angle to the inverse hyperbolic tangent

ATanh Calculator

Open Interval Domain

The ATanh(x) or inverse hyperbolic tangent shows inverse sigmoid behavior with domain (-1, 1).

Value must be between -1 and 1 (exclusive)
Results
Angle:

ATanh Function Curve

Curve of the ATanh function

The ATanh function has vertical asymptotes at x = ±1 and passes through origin.
Domain: (-1, 1), Range: ℝ

Inverse Sigmoid Behavior of ATanh

The inverse hyperbolic tangent function exhibits characteristic inverse sigmoid properties:

  • Domain: (-1, 1) (open interval)
  • Range: ℝ (all real numbers)
  • Zero Point: ATanh(0) = 0
  • Inverse: Tanh(ATanh(x)) = x for |x| < 1
  • Symmetry: Odd function ATanh(-x) = -ATanh(x)
  • Asymptotes: Vertical at x = ±1

Logarithmic Representation of ATanh Function

The inverse hyperbolic tangent function is expressed through logarithmic functions:

Basic Formula
\[\text{ATanh}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\]

Natural logarithm expression for |x| < 1

Inverse Relation
\[\tanh(\text{ATanh}(x)) = x\]

For |x| < 1

Formulas for the ATanh Function

Definition
\[\text{ATanh}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\]

Natural logarithm expression for |x| < 1

Inverse Relation
\[\tanh(\text{ATanh}(x)) = x \text{ for } |x| < 1\] \[\text{ATanh}(\tanh(y)) = y \text{ for all } y \in \mathbb{R}\]

Inverse relationship with restricted domain

Derivative
\[\frac{d}{dx} \text{ATanh}(x) = \frac{1}{1 - x^2}\]

First derivative for |x| < 1

Symmetry Property
\[\text{ATanh}(-x) = -\text{ATanh}(x)\]

Odd function (antisymmetric)

Asymptotic Behavior
\[\lim_{x \to 1^-} \text{ATanh}(x) = +\infty\] \[\lim_{x \to -1^+} \text{ATanh}(x) = -\infty\]

Vertical asymptotes at domain boundaries

Special Values

Important Values
ATanh(0) = 0 ATanh(0.5) ≈ 0.549 ATanh(-0.5) ≈ -0.549
Boundary Behavior
\[x = \pm 1: \pm\infty\] \[|x| \geq 1: \text{NaN}\]

Diverges at boundaries, undefined outside

Zero Point
\[\text{ATanh}(0) = 0\]

Function passes through origin

Properties
  • Inverse sigmoid function
  • Strictly increasing
  • Odd function symmetry
  • Open interval domain
Fisher Transform
\[\text{ATanh}(r) = \frac{1}{2}\ln\left(\frac{1+r}{1-r}\right)\]

Used in correlation analysis

Applications

Statistics (Fisher transform), machine learning, signal processing, neural networks.

Detailed Description of the ATanh Function

Definition and Input

The inverse hyperbolic tangent function ATanh(x) is the inverse function of the hyperbolic tangent. It exhibits characteristic inverse sigmoid behavior with vertical asymptotes at the domain boundaries.

Critical Property: The ATanh function requires |x| < 1 and diverges at x = ±1!
Input

The argument must be a number between -1 and 1 (exclusive). If the value is -1 or 1, the result is ±∞ (infinity). For values outside the range, the result is NaN.

Result

The result is given in degrees (full circle = 360°) or radians (full circle = 2π). The unit of measurement used is set using the Degrees or Radians menu.

Using the Calculator

Enter a value between -1 and 1 (exclusive). The ATanh function calculates the angle whose hyperbolic tangent equals the input value.

Mathematical Properties

Function Properties
  • Domain: (-1, 1) (open interval)
  • Range: ℝ (all real numbers)
  • Zero Point: ATanh(0) = 0
  • Symmetry: Odd function ATanh(-x) = -ATanh(x)
Inverse Sigmoid Properties
  • Inverse of bounded sigmoid Tanh function
  • Vertical asymptotes at x = ±1
  • Rapid growth near domain boundaries
  • S-shaped curve rotated 90° (inverse sigmoid)
Applications
  • Statistics: Fisher transformation for correlations
  • Machine Learning: Inverse sigmoid activation
  • Signal Processing: Bounded signal inversions
  • Neural Networks: Inverse activation functions
Practical Notes
  • Domain restriction: |x| < 1 (diverges at ±1)
  • Origin passage: ATanh(0) = 0
  • Odd function: ATanh(-x) = -ATanh(x)
  • Inverse relation: Tanh(ATanh(x)) = x for |x| < 1

Calculation Examples

Valid Values

ATanh(0) = 0

ATanh(0.5) ≈ 0.549

ATanh(0.9) ≈ 1.472

Negative Values

ATanh(-0.5) ≈ -0.549

ATanh(-0.9) ≈ -1.472

ATanh(-0.99) ≈ -2.647

Boundary Behavior

x → 1⁻: ATanh(x) → +∞

x → -1⁺: ATanh(x) → -∞

|x| ≥ 1: NaN

Statistical and Machine Learning Applications

Fisher Transformation

Correlation Analysis:

z = ½ln((1+r)/(1-r)) = ATanh(r)

Transforms correlations to normal distribution

Application: Statistical hypothesis testing for correlations.

Neural Networks

Inverse Activation:

x = ATanh(y) for bounded outputs

Inverse sigmoid transformations

Example: Inverse transformations in gradient computations.

Important Mathematical Relationships

Inverse Function Properties
\[\tanh(\text{ATanh}(x)) = x \text{ for } |x| < 1\] \[\text{ATanh}(\tanh(y)) = y \text{ for all } y \in \mathbb{R}\]

Inverse Sigmoid: Maps bounded interval to unbounded range.

Calculus Properties
\[\frac{d}{dx}\text{ATanh}(x) = \frac{1}{1-x^2}\] \[\int \text{ATanh}(x) dx = x\text{ATanh}(x) + \frac{1}{2}\ln(1-x^2) + C\]

Derivative: Grows rapidly near x = ±1.


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Special functions

Airy • Derivative Airy • Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye • Spherical-Bessel-J • Spherical-Bessel-Y • Hankel • Beta • Incomplete Beta • Incomplete Inverse Beta • Binomial Coefficient • Binomial Coefficient Logarithm • Erf • Erfc • Erfi • Erfci • Fibonacci • Fibonacci Tabelle • Gamma • Inverse Gamma • Log Gamma • Digamma • Trigamma • Logit • Sigmoid • Derivative Sigmoid • Softsign • Derivative Softsign • Softmax • Struve • Struve table • Modified Struve • Modified Struve table • Riemann Zeta

Hyperbolic functions

ACosh • ACoth • ACsch • ASech • ASinh • ATanh • Cosh • Coth • Csch • Sech • Sinh • Tanh

Trigonometrische Funktionen

ACos • ACot • ACsc • ASec • ASin • ATan • Cos • Cot • Csc • Sec • Sin • Sinc • Tan • Degree to Radian • Radian to Degree