arctanab

Arctanab is a term used to denote the arctangent of the product of two real numbers a and b, written as arctan(ab). It can be viewed as the composition arctan ∘ (multiplication by b) or, more generally, as the two-variable function f(a,b) = arctan(ab).

Definition and notation

For real a and b, arctanab := arctan(ab), where arctan is the inverse tangent function with range (−π/2, π/2). The function is defined for all real inputs ab.

Domain, range and basic properties

Since ab can take any real value, the domain of arctanab is all pairs (a,b) in R^2. The range of arctan is (−π/2, π/2). The function is continuous and infinitely differentiable in R^2. It is symmetric in a and b: arctan(ab) = arctan(ba).

Calculus

Partial derivatives are f(a,b) = arctan(ab) and satisfy ∂f/∂a = b / (1 + (ab)^2) and ∂f/∂b = a / (1 + (ab)^2). For fixed b > 0, f is increasing in a; for b < 0, f is decreasing in a. As |ab| grows, arctan(ab) tends to ±π/2 with the sign of ab.

Series and approximations

Around the origin, arctan(z) has the standard series arctan(z) = z − z^3/3 + z^5/5 − … for |z| ≤ 1. Thus arctan(ab) ≈ ab − (ab)^3/3 + (ab)^5/5 − …, valid for |ab| ≤ 1, with convergence governed by the usual arctan series.

Relation to other identities

There is no simple closed form for arctan(ab) in terms of arctan(a) and arctan(b). However, the identity tan( arctan a + arctan b ) = (a + b) / (1 − ab) relates arctan to addition formulas, illustrating how arctan(ab) fits into the broader arctangent framework.