arctanwH

arctanwH is not a standard, widely recognized mathematical term with a single agreed-upon definition. The exact meaning of the notation depends on the context in which it appears, and authors may use it to denote different ideas. The two most plausible interpretations are discussed below.

Interpretation A treats arctanwH as arctan(wH), the arctangent of the product wH. In this reading, f(w,H) = arctan(wH) maps pairs of real numbers to the interval (-pi/2, pi/2). The domain is all real w and H, and the function is continuous and differentiable. Since arctan is increasing, f increases with the product wH: for fixed H, f grows with w, and for fixed w, f grows with H. The partial derivatives are ∂f/∂w = H / (1 + (wH)^2) and ∂f/∂H = w / (1 + (wH)^2). As |wH| becomes large, f approaches ±pi/2.

Interpretation B treats arctanwH as H · arctan(w), the product of H with the arctangent of w. Here f(w,H) = H arctan(w). The domain is all real w and H, and the range is unbounded as H varies. For fixed H, f is monotone in w; for fixed w, f is linear in H. The partial derivatives are ∂f/∂w = H / (1 + w^2) and ∂f/∂H = arctan(w).

In practice, the exact meaning should be clarified by the source. Both interpretations rely on the standard arctan function and yield different algebraic and geometric properties. See also arctan, arctan2, and tangent.