tanh0

tanh0 is a shorthand reference to the value of the hyperbolic tangent function at zero, written mathematically as tanh(0). The hyperbolic tangent tanh(x) is defined by the formula tanh(x) = (e^x − e^(−x)) / (e^x + e^(−x)).

Evaluating at zero gives tanh(0) = 0. The function is odd, strictly increasing, and maps the real line onto the interval (−1, 1). As x approaches positive infinity, tanh(x) tends to 1, and as x approaches negative infinity, tanh(x) tends to −1. The derivative at any x is tanh′(x) = sech^2(x) = 1 − tanh^2(x); in particular, tanh′(0) = 1, giving a slope of 1 at the origin. The Maclaurin series expansion around x = 0 begins with tanh(x) = x − x^3/3 + 2x^5/15 − 17x^7/315 + ….

Tanh is related to other functions through identities such as tanh(x) = (e^(2x) − 1)/(e^(2x) + 1) and tanh(x) = 2·sigmoid(2x) − 1, where the sigmoid function is 1/(1 + e^(−2x)). In applied contexts, tanh serves as a smooth, saturating activation function in neural networks and appears in solutions to certain differential equations. The value tanh0 = tanh(0) equals 0.