arctan2im

arctan2im is a function name that appears in some mathematical software and libraries as a variant of the two-argument arctangent function for complex-valued inputs. It is not a standard mathematical constant or universally defined function, and its exact definition can vary by implementation. In practice, arctan2im is used to compute an angle or phase from a pair of real or complex inputs while attempting to preserve quadrant information in a way analogous to the arctan2 function for real numbers.

In several contexts, arctan2im(a, b) is defined in terms of the complex arctangent or via a division z = a + i b treated as a complex number. A common convention is that arctan2im(a, b) returns the principal value of the complex arctangent of a/b, with branch choices chosen to reflect the signs of a and b. If a and b are real, the function can reduce to the standard arctan2(y, x) = atan2(y, x), yielding a real angle in the range (-π, π].

Properties of arctan2im depend on the chosen convention and the underlying complex-arctan definition. It may yield real results for real inputs and potentially complex results for certain complex arguments. Implementations must handle edge cases such as b = 0 and a = 0 with well-defined limits, and numerical considerations include avoiding cancellation and maintaining stable branching.

Applications of arctan2im appear in areas requiring a two-argument arctangent for complex quantities, such as certain signal-processing pipelines, robotics computations, or symbolic algebra systems. The function is closely related to atan2, arctan, the complex logarithm, and the identity arctan z = (i/2) log((1 - i z)/(1 + i z)).