Complex Multiplication and Division

📐 Definition

Write $z_j = r_j e^{i\theta_j}$ with $r_j \ge 0$ and $\theta_j \in \mathbb{R}$ . Then

z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}

\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}, \qquad z_2 \ne 0

⚙️ Key Properties

Magnitude and Phase Composition

In polar form, multiplication multiplies magnitudes and adds phases; division divides magnitudes and subtracts phases (modulo $2\pi$ ):

|z_1 z_2| = |z_1|\,|z_2|, \qquad \operatorname{Arg}(z_1 z_2) \equiv \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \pmod{2\pi}

Distributivity (Rectangular Form)

Complex multiplication is distributive over addition:

z(a+b) = za + zb

Conjugation Compatibility

\overline{z_1 z_2} = \overline{z_1}\,\overline{z_2}, \qquad \overline{\frac{z_1}{z_2}} = \frac{\overline{z_1}}{\overline{z_2}}

🎯 Special Cases and Limits

Division by zero is undefined. For purely real numbers, the formulas reduce to real multiplication/division with phase restricted to $0$ or $\pi$ (principal value).

Polar–Cartesian Transforms Coordinate conversions that simplify multiplication/division

Complex Magnitude Magnitudes multiply under products

Complex Phase Phases add/subtract under multiplication/division

Usage in Oakfield

Oakfield uses complex multiplication/division throughout its complex-field operators, especially in spectral transforms and modulation-style mixing:

FFT-based operators (fft_plan plus operators like fft_convert, linear_dissipative, dispersion) multiply complex buffers by twiddles, gains, and phase factors (e.g. dispersion applies per-bin factors $e^{i\phi(k)}$ via cexp).
Mixer complex modes use multiplication heavily: MULTIPLY/RING_MOD are direct products, FM applies $L\cdot e^{iR}$ , and PM applies $|L|\cdot e^{i(\operatorname{Arg}(L)+R)}$ .
Polar/phase-preserving paths normalize by magnitude using division ( $z/|z|$ ) to preserve phase while changing amplitude (e.g. analytic_warp and remainder in polar mode, and phase_feature extraction).
Low-level helpers exist for explicit SimComplexDouble math (sim_complex_mul, sim_complex_div), though many kernels also use C99 double complex directly.

Historical Foundations

📜 Geometric Interpretation

Once complex numbers are identified with planar vectors, multiplication corresponds to a combined scaling and rotation; this viewpoint explains why polar coordinates make multiplication and division particularly transparent.
🌍 Modern Perspective

Polar-form composition remains central in signal processing and spectral methods, where gain and phase are manipulated directly.

📚 References

Needham, Visual Complex Analysis
Ahlfors, Complex Analysis