Sqrtht

Sqrtht is a compact notation used in mathematical exposition to denote a specific composite function formed by taking the square root of the hyperbolic tangent of its argument. In this article, sqrtht is defined for real inputs by the rule sqrtht(x) = sqrt(tanh(x)). Because the square root is interpreted in the real sense, the function is defined only where tanh(x) ≥ 0, which occurs for x ≥ 0. Extensions to complex inputs are possible via analytic continuation, but such extensions are beyond the scope of basic real analysis.

Notationally, sqrtht(x) combines two well-known functions: the hyperbolic tangent, which maps the real line to the interval (-1, 1), and the principal square root, which requires a nonnegative argument. Consequently, the real-valued domain of sqrtht is [0, ∞), with tanh(0) = 0 giving sqrtht(0) = 0 and tanh(x) → 1 as x → ∞ giving sqrtht(x) → 1.

Key properties include monotonicity and boundedness: the function is increasing on [0, ∞) and its range is [0, 1). It is differentiable on (0, ∞) with derivative sqrtht'(x) = sech^2(x) / (2 sqrt(tanh(x))). At x = 0 the derivative is not defined in the real-valued sense due to the sqrt(tanh(x)) in the denominator.

Examples: sqrtht(0) = 0; sqrtht(1) = sqrt(tanh(1)) ≈ sqrt(0.7616) ≈ 0.872; as x grows, sqrtht(x) approaches 1.

In usage, sqrtht is primarily a didactic construct to illustrate composition, domain restrictions, and basic asymptotic behavior. See also: square root, hyperbolic tangent, function composition.