The Hyperbolic Tangent Function | Definition, Properties, and Applications

tanh x

The hyperbolic tangent function, denoted as tanh x, is a mathematical function that operates on a real number x

The hyperbolic tangent function, denoted as tanh x, is a mathematical function that operates on a real number x. It is defined as the ratio of the hyperbolic sine function (sinh x) to the hyperbolic cosine function (cosh x):

tanh x = sinh x / cosh x

In terms of exponential functions, tanh x can be expressed as:

tanh x = (e^x – e^(-x)) / (e^x + e^(-x))

The hyperbolic tangent function has a range from -1 to 1, which means its values can go negative, reach zero, or remain positive. It is an odd function, symmetric about the origin, and its graph resembles an elongated “S” shape.

The tanh function has several applications, particularly in mathematical and computational fields. It is commonly used in neural networks, where it serves as an activation function for neurons. Additionally, it finds utility in solving integrals and differential equations involving hyperbolic trigonometric functions.

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