Understanding the Hyperbolic Tangent Function

What is the Hyperbolic Tangent Function?

The hyperbolic tangent function, denoted as tanh(x), is one of the fundamental hyperbolic functions in mathematics. It is analogous to the trigonometric tangent function but is defined in terms of exponential functions rather than angles. The hyperbolic tangent function has numerous applications in various fields, including neural networks, signal processing, and control systems.

Formula and Definition

The hyperbolic tangent function is defined as:

\[\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\]

Where:

• \(x\) is any real number
• \(e\) is Euler's number (approximately 2.71828)
• \(\sinh(x)\) is the hyperbolic sine function
• \(\cosh(x)\) is the hyperbolic cosine function

Properties of tanh(x)

• Domain: All real numbers
• Range: (-1, 1) (all real numbers between -1 and 1, exclusive)
• tanh(x) is an odd function: tanh(-x) = -tanh(x)
• tanh(0) = 0
• As x approaches positive infinity, tanh(x) approaches 1
• As x approaches negative infinity, tanh(x) approaches -1
• The graph of tanh(x) is symmetric about the origin

Calculation Steps

1. Input any real number x.
2. Calculate \(e^x\) and \(e^{-x}\).
3. Calculate the numerator: \(e^x - e^{-x}\).
4. Calculate the denominator: \(e^x + e^{-x}\).
5. Divide the numerator by the denominator.

Example Calculation

Let's calculate tanh(1):

1. Input: x = 1
2. Calculate: \(e^1 \approx 2.71828\) and \(e^{-1} \approx 0.36788\)
3. Numerator: \(2.71828 - 0.36788 = 2.35040\)
4. Denominator: \(2.71828 + 0.36788 = 3.08616\)
5. Divide: \(2.35040 / 3.08616 \approx 0.76159\)

Therefore, tanh(1) ≈ 0.76159

Visual Representation

This graph illustrates the hyperbolic tangent function. The point (1, 0.76) corresponds to tanh(1).