arctangent

Arctangent, denoted arctan or tan^{-1}, is the inverse function of the tangent function when tangent is restricted to its principal branch on the interval (-pi/2, pi/2). For any real x, arctan x is the unique angle in (-pi/2, pi/2) whose tangent equals x. The function arctan: R -> (-pi/2, pi/2) is strictly increasing, with arctan(0) = 0, and limits arctan x approaching pi/2 as x → ∞ and -pi/2 as x → -∞.

Key properties include its derivative and limits: d/dx arctan x = 1/(1 + x^2), and as x grows without bound, arctan x tends to pi/2; as x decreases without bound, arctan x tends to -pi/2. The principal value of the inverse tangent makes it the inverse of tan on (-pi/2, pi/2); because tan is periodic with period pi, arctan is not the inverse of tan on all real inputs without restricting the domain.

Series and computation: arctan x has the Taylor (Mercator) series arctan x = x - x^3/3 + x^5/5 - x^7/7 + ... for |x| ≤ 1, with the endpoints x = ±1 giving arctan(±1) = ±pi/4.

Applications and extensions: arctan is widely used to determine angles from slopes, in integration and trigonometry, and appears in complex analysis with the common extension arctan z = (i/2)[ln(1 − i z) − ln(1 + i z)].