Arctanbh

Arctanbh is a compact notation used in some contexts to denote a two-argument function defined by arctanbh(b, h) = arctan(bh), i.e., the inverse tangent of the product of its two inputs. In this sense, arctanbh simply applies the standard arctan to the real number obtained by multiplying b and h.

Domain and range are straightforward for real inputs. For all real b and h, the product bh is real, and since arctan is defined for all real arguments, arctanbh(b, h) is defined for all pairs (b, h) ∈ R^2. The range of arctan is (-π/2, π/2), so arctanbh also takes values only in this interval.

Basic properties follow from the underlying arctan function. For fixed h, the partial derivative with respect to b is ∂/∂b arctan(bh) = h / (1 + (bh)²). Similarly, ∂/∂h arctan(bh) = b / (1 + (bh)²). The function is symmetric in b and h since the product bh is symmetric: arctanbh(b, h) = arctanbh(h, b). Special cases include arctanbh(b, 1) = arctan(b), arctanbh(1, h) = arctan(h), and arctanbh(0, h) = 0.

Relation to other functions is direct: arctanbh reduces to the ordinary arctan when one argument is unity, and it shares all the standard numerical properties of the arctan function, including its behavior for large or small inputs (as |bh| → ∞, arctan(bh) → ±π/2).

Notes: arctanbh is not a standard widely adopted notation, and its meaning may vary by author. When used, it is typically clear from context that it represents arctan of a product of two real quantities.