Complex Phase

📐 Definition

For a complex number $z = x + iy \neq 0$, the complex phase (argument) is the angle of $z$ in the complex plane. The principal value is commonly taken in $(-\pi,\pi]$:

$$\operatorname{Arg}(z) = \operatorname{atan2}(y,x) \in (-\pi,\pi]$$

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For $z=0$, neither $\operatorname{Arg}(z)$ nor $\arg(z)$ is defined.

Domain & Codomain

Domain: $\mathbb{C}\setminus\{0\}$ (or phases of nonzero complex numbers); Codomain: principal argument $(-\pi,\pi]$ and multivalued classes modulo $2\pi$.

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⚙️ Key Properties

Polar Decomposition

Away from the branch cut used to define the principal value, a complex number can be written as

$$z = |z|\,e^{i\operatorname{Arg}(z)}$$

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Multiplication and Division (Modulo $2\pi$)

$$\operatorname{Arg}(z_1 z_2) \equiv \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \pmod{2\pi}$$

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$$\operatorname{Arg}\!\left(\frac{z_1}{z_2}\right) \equiv \operatorname{Arg}(z_1) - \operatorname{Arg}(z_2) \pmod{2\pi}, \qquad z_2 \ne 0$$

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Branch Cut and Discontinuity

Any single-valued choice of argument (such as $\operatorname{Arg}$) introduces a branch cut (commonly the negative real axis). Crossing the cut produces a jump of $2\pi$ in the principal value.

Relation to the Complex Logarithm

For the principal complex logarithm,

$$\operatorname{Log}(z) = \ln|z| + i\operatorname{Arg}(z)$$

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🎯 Special Cases and Limits

$z$	$\operatorname{Arg}(z)$ (principal)
$z>0$ real	$0$
$z<0$ real	$\pi$
$z=iy,\ y>0$	$\tfrac{\pi}{2}$
$z=iy,\ y<0$	$-\tfrac{\pi}{2}$
$z=0$	undefined

Continuous phase tracking typically requires unwrapping, replacing principal values by a continuous representative.

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🔗 Related Functions

Complex Magnitude Radial component used with phase

Polar–Cartesian Transforms Coordinate conversions involving phase

Logarithm Relation via complex log: Log(z) = ln|z| + i Arg(z)

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Usage in Oakfield

Applications in Oakfield

In Oakfield, phase is used both for visualization and for phase-aware operator behavior:

• GPU phase visualization computes $\theta=\operatorname{atan2}(\Im z,\Re z)$ and maps it to hue (the “phase” view shader).
• Mixer modulation modes treat phase explicitly: FM applies $L \mapsto L\cdot e^{iR}$ and PM applies $L \mapsto |L|\cdot e^{i(\operatorname{Arg}(L)+R)}$ (implemented with cexp/carg).
• Phase coherence diagnostics typically avoid explicitly forming $\operatorname{Arg}(z)$ and instead work with normalized phasors $z/|z|$ (see phase_coherence metrics in runtime field stats).
• Stimulus rotations for complex fields are applied as unit-magnitude complex multipliers (equivalently, adding a phase offset).

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Historical Foundations

1. 📜 Complex Plane Representation

   The interpretation of complex numbers as points in the plane enabled a geometric notion of angle (argument), developed in late 18th–early 19th century work by Wessel and Argand and incorporated into the foundations of complex analysis.

2. 🌍 Modern Perspective

   In modern computation, $\arg(z)$ is typically implemented via atan2 and treated together with branch cuts and unwrapping when continuity in time or space is required.

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📚 References

• Ahlfors, Complex Analysis
• Conway, Functions of One Complex Variable