atan2Imz

atan2Imz is a mathematical function used to determine the phase angle of a complex number by applying the two-argument arctangent to its imaginary and real parts. In this sense, atan2Imz(z) can be viewed as the principal value of the argument of z = x + i y, computed as atan2(y, x). The name highlights the role of the imaginary part as the first argument and the real part as the second.

Definition and domain: For real x and y with not both zero, atan2Imz(y, x) = atan2(y, x). When z = 0 (i.e., x = 0 and y = 0), the value is undefined in the standard formulation; some implementations return a convention such as 0 or NaN.

Relation to atan2 and to Im/Re: atan2Imz essentially mirrors the standard atan2 function used in two-argument arctangent computations, but framed in the context of complex numbers. It treats the input pair as Im(z) and Re(z), yielding the argument (angle) of z in the complex plane.

Properties: The result lies in the range (-π, π], with the exact quadrant determined by the signs of x and y. The function is discontinuous along the negative real axis when crossing the branch cut, and values outside the principal range can be adjusted by adding or subtracting 2π as needed for phase unwrapping.

Applications: It is used to compute the phase of complex-valued signals, in plotting and analysis of complex fields, and in algorithms that require a quadrant-aware angle from a complex input, such as phasor representation and quadrature demodulation.

Numerical considerations: Use robust implementations of atan2 to handle edge cases where x and/or y are near zero. Define behavior at z = 0 consistently within the surrounding application as part of error handling or by convention. See also atan2, arg, and complex number representations.