ArgZ1

argZ1 is a theoretical extension of the complex argument function in complex analysis, defined as a family of branch-specific angle mappings parameterized by Z1. It serves to encode the angular component of a complex number with an explicitly chosen branch.

Definition: For nonzero z in the complex plane, write z = r e^{iθ} with r > 0. The value argZ1(z) is the principal argument θ restricted to an interval I(Z1) of length 2π, where I(Z1) is determined by a branch cut along a ray from the origin specified by the parameter Z1. If z lies on the branch cut, argZ1(z) is undefined; otherwise argZ1(z) is unique. The function is related to the complex logarithm by argZ1(z) = Im(LogZ1(z)).

Properties: On the complex plane minus the branch cut, argZ1 is single-valued and discontinuous across the cut. It increases by 2π when encircling the origin once without crossing the cut. It is compatible with the standard argument when Z1 selects a conventional cut, such as the negative real axis.

Computation: In Cartesian coordinates, argZ1(x+iy) can be computed as θ0 = atan2(y, x) and then adjusted by adding or subtracting multiples of 2π to fit I(Z1). The choice of Z1 determines the offset of the interval.

Applications and limitations: The concept appears in theoretical discussions of phase representations, polar transforms, and phase unwrapping in signal processing contexts. It emphasizes that the angular value depends on the chosen branch and that a single-valued version requires a branch cut, which introduces a discontinuity along the cut.

See also: Argument (complex analysis), complex logarithm, atan2 function.