arctanm

arctanm is the inverse tangent function applied to a real input m, commonly written arctan(m). It returns the unique angle θ in the interval (-π/2, π/2) such that tan(θ) = m. In this sense, arctanm serves as the principal value of the arctangent function and is widely used in mathematics, physics, and engineering.

Domain and range: The input m can be any real number, and the output arctanm lies in (-π/2, π/2). The function is continuous and strictly increasing on its domain.

Key properties: arctanm is an odd function, meaning arctanm(-m) = -arctanm(m). Its derivative with respect to m is 1/(1 + m^2). The limits at infinity are arctanm(m) → π/2 as m → ∞ and arctanm(m) → -π/2 as m → -∞. The fundamental identity tan(arctanm) = m holds, and arctanm(tan(θ)) equals θ for θ in (-π/2, π/2).

Series and identities: For |m| ≤ 1, arctanm = m − m^3/3 + m^5/5 − … (the alternating series). Addition formulas express arctanm(a) + arctanm(b) in terms of arctanm((a+b)/(1−ab)) with attention to quadrant adjustments.

Notation and usage: While arctanm is sometimes used informally to emphasize the variable m, most texts write arctan(m). The function is fundamental in converting slopes to angles, polar coordinate conversions, and various computational algorithms requiring inverse tangent values.