tanhsqrt2

tanhsqrt2 refers to the value of the hyperbolic tangent function at the square root of 2. In mathematical notation, this is tanh(sqrt(2)). The hyperbolic tangent is defined by tanh x = (e^x − e^−x)/(e^x + e^−x) or equivalently tanh x = (e^{2x} − 1)/(e^{2x} + 1). For x = sqrt(2), tanh(sqrt(2)) = (e^{2 sqrt(2)} − 1)/(e^{2 sqrt(2)} + 1).

Numerically, sqrt(2) ≈ 1.41421356, so e^{2 sqrt(2)} ≈ 16.9123, which gives tanh(sqrt(2)) ≈ 0.8884. More precise evaluations yield tanh(sqrt(2)) ≈ 0.888344… depending on the precision of the exponential calculations.

Properties and context: tanh(sqrt(2)) is a real number strictly between 0 and 1, since tanh x is increasing on the positive real axis and tends to 1 as x grows. There is no known simple closed form for this value in terms of elementary constants. It is typically computed via standard numerical routines for the exponential function or directly via the hyperbolic tangent function.

In applications, tanh(sqrt(2)) serves as an example of evaluating hyperbolic functions at irrational arguments and illustrates how the function maps positive inputs to values in (0, 1). See also hyperbolic functions, square root of 2, and numerical evaluation of transcendental functions.