Inverse Tanh Calculator

Inverse Hyperbolic Tangent Formula:

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

Result (x):

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1. What is Inverse Hyperbolic Tangent?

The inverse hyperbolic tangent (artanh or tanh⁻¹) is the inverse function of the hyperbolic tangent function. It returns the value whose hyperbolic tangent is the given number.

2. How Does the Calculator Work?

The calculator uses the inverse hyperbolic tangent formula:

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

Where:

\( y \) — Input value (must be between -1 and 1, exclusive)
\( x \) — Result in radians
\( \ln \) — Natural logarithm

Explanation: The function is undefined at y = ±1 and grows rapidly as y approaches these limits.

3. Applications of Inverse Tanh

Details: The inverse hyperbolic tangent is used in various fields including physics (special relativity), engineering (signal processing), and statistics (Fisher transformation).

4. Using the Calculator

Tips: Enter a value between -1 and 1 (not including -1 or 1). The calculator will return the corresponding inverse hyperbolic tangent value in radians.

5. Frequently Asked Questions (FAQ)

Q1: Why is the input restricted to -1 < y < 1?
A: The hyperbolic tangent function only outputs values in this range, so its inverse is only defined for these inputs.

Q2: What happens at y = ±1?
A: The function approaches ±∞ as y approaches ±1, making these points undefined.

Q3: How is this related to the regular inverse tangent?
A: While both are inverse trigonometric functions, tan⁻¹ is the inverse of circular tangent, while artanh is the inverse of hyperbolic tangent.

Q4: Can I calculate this for complex numbers?
A: Yes, but this calculator only handles real-valued inputs between -1 and 1.

Q5: What are some practical uses of this function?
A: It's used in special relativity for rapidity calculations, in statistics for variance stabilization, and in digital signal processing.