arctanx

Arctan x, written arctan x, is the inverse function of the tangent function restricted to the interval (-π/2, π/2). It returns the unique angle θ in (-π/2, π/2) whose tangent equals x, i.e., tan θ = x. The domain of arctan is all real numbers, and its range is (-π/2, π/2). The function is continuous and strictly increasing, and it is odd: arctan(-x) = -arctan x.

Key calculus and analytic properties include its derivative and integral representations. The derivative of arctan x is 1/(1+x^2). A common integral representation is arctan x = ∫ from 0 to x of dt/(1+t^2). An antiderivative is ∫ arctan x dx = x arctan x − (1/2) ln(1+x^2) + C.

Among its series representations, the Maclaurin series arctan x = x − x^3/3 + x^5/5 − x^7/7 + … converges for |x| ≤ 1, with the endpoint values giving arctan 1 = π/4 and arctan(−1) = −π/4. The series provides practical approximations for small x.

Special values and limits are straightforward: arctan 0 = 0, arctan 1 = π/4, arctan −1 = −π/4. As x → ∞, arctan x → π/2, and as x → −∞, arctan x → −π/2.

In trigonometric identities, arctan appears in sum formulas such as arctan x + arctan y = arctan((x+y)/(1−xy)) with an appropriate adjustment by ±π to account for the correct quadrant. In polar coordinates, arctan2(y, x) generalizes arctan to determine the angle of a vector (x, y) with quadrant information.