arctanm2

Arctanm2 denotes the mathematical function f(m) = arctan(m^2), where arctan is the inverse tangent. Here m is a real variable. The notation emphasizes squaring the input before applying the arctangent.

Domain and range: the argument m^2 is nonnegative for all real m, so f maps the real numbers to the interval [0, π/2). The function is even: f(-m) = f(m). As m grows without bound, f(m) increases toward π/2 but never equals it.

Differentiation and inverse: Differentiation yields f'(m) = 2m / (1 + m^4). This makes f strictly increasing on the nonnegative axis and symmetric about the origin. The inverse relation, for m ≥ 0, is m = sqrt(tan y) with y ∈ [0, π/2).

Special values and expansion: f(0) = 0 and f(1) = π/4. For small m, arctan(m^2) expands as m^2 − m^6/3 + m^10/5 − … .

Computational notes and applications: arctanm2 is typically computed as atan(m*m) in programming environments. Care is needed to avoid overflow for large m. The function serves as a saturating nonlinear mapping in various mathematical contexts and appears in analyses where a nonnegative input to arctangent is relevant.

---