3.2
Math Foundations
§3 · Complex exponentials and phasors

3.2 Damped oscillations as phasors

The phasors of 3.1 lived on the unit circle: eiθe^{i\theta} with real argument θ\theta has magnitude 1. This is the right object for steady-state oscillations at a single frequency. But real systems lose energy — to friction, to viscous drag, to resistance — and their oscillations decay over time. The complex-exponential machinery handles this without modification: we just let the argument become complex.

This lesson develops the complex-argument phasor e(−γ+iω)te^{(-\gamma + i\omega) t} and its spiral interpretation. It is the eigenfunction of every linear damped system on the bookshelf — the mathematical object behind the underdamped regime of Foundations 5.3, the cochlear filter of Hearing Ch 4, and the ring-down of every real resonator.

A complex argument: e(λ)te^{(\lambda) t} with λ\lambda complex

Take the real exponential eλte^{\lambda t} and let λ\lambda be a complex number: λ=−γ+iω\lambda = -\gamma + i\omega with γ≥0\gamma \geq 0. Then

eλt  =  e(−γ+iω)t  =  e−γt eiωt.e^{\lambda t} \;=\; e^{(-\gamma + i\omega) t} \;=\; e^{-\gamma t}\, e^{i\omega t}.

Using Euler’s formula on the second factor:

e(−γ+iω)t  =  e−γt [cos⁡(ωt)+isin⁡(ωt)].e^{(-\gamma + i\omega) t} \;=\; e^{-\gamma t}\, [\cos(\omega t) + i \sin(\omega t)].

This is a complex number whose magnitude is ∣e(−γ+iω)t∣=e−γt|e^{(-\gamma + i\omega) t}| = e^{-\gamma t} (shrinking with time) and whose angle is ωt\omega t (rotating with time). Plotted in the complex plane, it traces a logarithmic spiral inward.

The real part of this complex function is

Re⁡[e(−γ+iω)t]  =  e−γt cos⁡(ωt),\operatorname{Re}\bigl[ e^{(-\gamma + i\omega) t} \bigr] \;=\; e^{-\gamma t}\, \cos(\omega t),

which is a sinusoid at frequency ω\omega enclosed in a decaying exponential envelope ±e−γt\pm e^{-\gamma t}. This is the time-domain signature of an underdamped oscillator — the ringdown of a struck bell, the decay of a tuning fork, the impulse response of a band-pass filter.

The complex picture (spiral) and the real-time picture (damped cosine) are two faces of the same object.

ReImz(t) = e^((−γ + iω) t) — complex planeRe[z(t)] = cos(ωt) e^(−γt) — time domaint →t = 0.00, |z| = 1.000, arg z = 0°

A complex exponential with complex argument (−γ + iω) t traces a logarithmic spiral in the complex plane — the magnitude e−γt shrinks while the angle ωt rotates. The real part, plotted on the right, is a sinusoid at frequency ω inside an exponentially-decaying envelope ±e−γt (red dashed). This is the eigenfunction of a damped harmonic oscillator: z(t) = eλt with λ = −γ + iω from the characteristic equation λ² + 2γλ + ω₀² = 0. The complex picture packages amplitude and phase into a single rotating arrow; the spiral and the damped cosine are two views of the same object.

Slide ω\omega (rotation rate) and γ\gamma (damping rate); watch the spiral on the left and the damped cosine on the right. Three things to feel for:

Where this comes from

The complex argument λ=−γ+iω\lambda = -\gamma + i\omega isn’t arbitrary. It is exactly what the characteristic equation of a damped oscillator produces. Recall (Foundations 5.3) that the damped oscillator ODE

x¨+2γ x˙+ω02 x  =  0\ddot x + 2\gamma\, \dot x + \omega_0^2\, x \;=\; 0

is solved by substituting x=eλtx = e^{\lambda t}, which gives the quadratic

λ2+2γ λ+ω02  =  0\lambda^2 + 2\gamma\, \lambda + \omega_0^2 \;=\; 0

with roots

λ±  =  −γ±γ2−ω02.\lambda_{\pm} \;=\; -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}.

When γ<ω0\gamma < \omega_0 (underdamped), the square root is imaginary and the roots are complex conjugates:

λ±  =  −γ±iωd,ωd≡ω02−γ2.\lambda_{\pm} \;=\; -\gamma \pm i \omega_d, \qquad \omega_d \equiv \sqrt{\omega_0^2 - \gamma^2}.

These complex eigenvalues are exactly the complex argument we’ve been visualising. The real part −γ-\gamma sets the spiral’s decay rate; the imaginary part ωd\omega_d sets its rotation rate (the damped natural frequency, slightly lower than the undamped ω0\omega_0 because the damping “drags” the oscillation).

The general solution of the damped-oscillator ODE is therefore a sum of two complex-conjugate spirals, Aeλ+t+Beλ−tA e^{\lambda_+ t} + B e^{\lambda_- t}. Combining the conjugate pair into a real function gives the damped sinusoid

x(t)  =  e−γt [ Ccos⁡(ωdt)+Dsin⁡(ωdt) ].x(t) \;=\; e^{-\gamma t}\, [\,C \cos(\omega_d t) + D \sin(\omega_d t)\,].

The complex-exponential representation packages this whole story into one object, e(−γ+iωd)te^{(-\gamma + i\omega_d) t}, with the spiral picture above. That is what makes the complex-exponential ansatz so powerful.

A complex-eigenvalue dictionary

The position of λ\lambda in the complex plane fully determines the qualitative behaviour of eλte^{\lambda t}:

λ\lambda positionBehaviour of eλte^{\lambda t}
Real, negativeExponential decay (no oscillation)
Real, positiveExponential growth (unstable)
Real, zeroConstant
Pure imaginarySteady oscillation at frequency $
Complex, Re λ<0\mathrm{Re}\,\lambda < 0Decaying oscillation (logarithmic spiral inward)
Complex, Re λ>0\mathrm{Re}\,\lambda > 0Growing oscillation (logarithmic spiral outward)

This dictionary is the whole content of linear-stability analysis. Eigenvalues in the left half-plane (negative real part) mean stable; in the right half-plane mean unstable; on the imaginary axis mean marginal. Control engineers and circuit designers spend their lives keeping eigenvalues on the correct side of that vertical line. The geometric picture from Foundations 5.3 — characteristic roots in the complex plane — is exactly the spiral picture of this lesson, viewed from the complex-eigenvalue side.

What this is useful for

Beyond the obvious (the damped oscillator itself), the complex-eigenvalue picture is central to:

The next lesson, 3.3 — Plane waves and complex impedance, extends the phasor framework to fields that depend on both space and time, producing plane waves and the impedance-in-the-complex-plane representation used throughout the Sound book.