tanh

Tanh refers to the hyperbolic tangent function. For real numbers x it is defined by tanh x = (e^x − e^(−x)) / (e^x + e^(−x)); equivalently tanh x = sinh x / cosh x. The function extends to a meromorphic function on the complex plane, with poles at z where cosh z = 0 (for example z = iπ/2 + iπk).

Tanh is an odd, strictly increasing function that maps the real line onto the interval (−1, 1). As x → ∞, tanh x → 1; as x → −∞, tanh x → −1. Its derivative is sech^2 x, where sech x = 1 / cosh x. The inverse function is artanh, defined on (−1, 1) by artanh y = 1/2 ln((1+y)/(1−y)).

Several equivalent expressions relate tanh to exponentials: tanh x = (e^{2x} − 1) / (e^{2x} + 1) = 2/(1 + e^(−2x)) − 1.

Series and asymptotics: the Maclaurin series begins tanh x = x − x^3/3 + 2x^5/15 − 17x^7/315 + …, so near zero tanh x ≈ x. For large |x|, tanh x approaches ±1 exponentially fast.

Applications: tanh is widely used as an activation function in neural networks, offering outputs in (−1, 1) and a zero-centered range. It also appears in physics and applied mathematics, including statistical mechanics and the solution of certain differential equations. Its properties as a smooth, bounded, monotone, and differentiable function underlie many theoretical and practical uses.