Understanding the Inverse Hyperbolic Tangent Function

What is the Inverse Hyperbolic Tangent Function?

The inverse hyperbolic tangent function, denoted as atanh(x) or tanh⁻¹(x), is the inverse function of the hyperbolic tangent. It returns the hyperbolic angle whose hyperbolic tangent is the input value. This function is important in various fields of mathematics, physics, and engineering, particularly in problems involving exponential growth or decay and in the theory of special relativity.

Formula and Definition

The inverse hyperbolic tangent function is defined as:

\[atanh(x) = \frac{1}{2}\ln(\frac{1+x}{1-x})\]

Where:

• \(x\) is any real number between -1 and 1 (exclusive)
• \(\ln\) is the natural logarithm function

Properties of atanh(x)

• Domain: -1 < x < 1
• Range: All real numbers
• atanh(0) = 0
• The graph of atanh(x) is symmetric about the origin
• atanh(x) is a strictly increasing function for -1 < x < 1
• As x approaches 1 or -1, atanh(x) approaches positive or negative infinity, respectively

Calculation Steps

1. Input any real number x where -1 < x < 1.
2. Calculate 1 + x and 1 - x.
3. Divide (1 + x) by (1 - x).
4. Calculate the natural logarithm of the result from step 3.
5. Multiply the result from step 4 by 1/2.

Example Calculation

Let's calculate atanh(0.5):

1. Input: x = 0.5
2. Calculate: 1 + 0.5 = 1.5 and 1 - 0.5 = 0.5
3. Divide: 1.5 / 0.5 = 3
4. Calculate: ln(3) ≈ 1.098612
5. Multiply: 1/2 * 1.098612 ≈ 0.549306

Therefore, atanh(0.5) ≈ 0.549306

Visual Representation

This graph illustrates the inverse hyperbolic tangent function. The point (0.5, 0.55) corresponds to atanh(0.5).