atan2Imzz

atan2Imzz is a term used in some mathematical and software contexts to denote a function that computes the argument (angle) of a complex number by applying the two-argument arctangent to its imaginary and real parts. Formally, for a complex number z = x + iy, atan2Imzz(z) is defined as atan2(Im(z), Re(z)) = atan2(y, x). This yields the principal value of the angle that the complex number makes with the positive real axis.

In essence, atan2Imzz is equivalent to the standard complex-argument function Arg(z). Depending on the implementation, it may be presented as a wrapper around the common atan2 function, with inputs derived from the real and imaginary components of z. The typical range is (-π, π], though some libraries may use different conventions for edge cases. The value is undefined at z = 0, and implementations may return NaN or a domain-specific sentinel.

Usage and context: The function is used to determine the orientation or phase of a complex quantity in fields such as signal processing, control systems, and computer graphics. It provides a robust way to compute angles across all quadrants without the discontinuities that arise from using arctan(y/x) alone.

Examples:

- z = 1 + i yields atan2(1, 1) = π/4 ≈ 0.7854.

- z = -1 + i yields atan2(1, -1) = 3π/4 ≈ 2.3562.

See also: Arg (complex analysis), atan2, complex number.