Chapter 6

The Map of s

How does a spinning number make a real wiggle, and what does every s look like?

Back to the spring (and a new mistake)

June: Last time we built a number that spins. Let's hand it to the spring and see if it behaves.

The spring needed $a^2 = -k/m = -4$ . Try $a = 2i$ : $(2i)^2 = 4\,i^2 = -4$ . ✓ And $a = -2i$ works too.

Theo: Fine, I was wrong. The guess-an-exponential trick survives. The spring is $e^{2it}$ , a spinner going round twice every… 2π seconds.

Mika: So the spring's position is $e^{2it}$ ? Right now it's at… $0.4 + 0.9i$ metres?

June: A spring can't be 0.9 metres sideways. It only moves up and down.

Mika forgot the other answer. We found two solutions, $e^{2it}$ and $e^{-2it}$ : a spinner and its mirror twin going the other way round. Add them up:

Chapter 6: The Map of s ↗

Mika: The sideways parts cancel perfectly! The sum just slides along the real line, back and forth.

x(t) = \tfrac12\left(e^{i\omega t} + e^{-i\omega t}\right) = \cos(\omega t),\qquad \omega = \sqrt{k/m}

Why is the twin always there? The spring's rule has only real numbers in it ( $m$ and $k$ , no $i$ ). Flip a solution top to bottom, turning every $i$ into $-i$ , and the rule can't tell the difference: the mirror image is a solution too. So for anything real the spinners come in mirror pairs, and together their sideways parts cancel, leaving a real wiggle we can measure.

The map of s

Chapter 4 gave us exponentials that grow or shrink. Now we have spinners that wiggle. Put them together: $e^{st}$ with $s = \sigma + i\omega$ .

e^{st} = e^{(\sigma + i\omega)t} = \underbrace{e^{\sigma t}}_{\text{grow/shrink}}\cdot\underbrace{e^{i\omega t}}_{\text{spin}}

So every number s is a recipe: σ (the along part) says how fast to grow or shrink, ω (the sideways part) says how fast to spin. Together they draw a spiral, and the spiral's shadow is the signal we'd see.

Predict first! Put s at $-0.5 + 3i$ . What does the shadow of $e^{st}$ look like?

Chapter 6: The Map of s ↗

Take a tour of the map by dragging s (or using the arrow keys):

Left half (σ < 0): the spiral shrinks, the wiggle dies away.
Right half (σ > 0): the spiral grows, the wiggle explodes.
On the vertical line (σ = 0): a perfect circle, a wiggle that goes on forever.
On the horizontal line (ω = 0): no spin at all, just the plain exponentials from Chapter 4.
Further from the horizontal line (up or down): faster spinning, faster wiggles. Further left: faster fading.

June: So a single dot on this map is a whole personality: how fast it wiggles and whether it calms down or blows up.

A quick detour: everything is made of spinners

One last surprise. Take spinners of different speeds (1, 3, 5, 7… times as fast), shrink each one and start it at just the right angle, then add their shadows. With enough of them you can build a huge range of shapes, even a square wave with sharp corners.

Chapter 6: The Map of s ↗

Fourier showed how many periodic signals can be assembled from spinning pieces. A square wave is one example: more pieces make its flat parts and sharp edges clearer. This is a powerful way to describe signals, with conditions and limits we can explore later.

What we know so far

What we figured out

Real wiggles come from mirror-twin spinners whose sideways parts cancel.
$e^{st}$ with $s = \sigma + i\omega$ spins and grows or shrinks: a spiral. Its shadow is a growing or dying wiggle.
The map of s: left calms down, right blows up, up means faster wiggles.
Many repeating signals, even a square wave, can be built by adding spinners of different speeds (Fourier). Right next to a jump the sum always overshoots by about 9%.

June: "So where on this map does our drone live?"

Chapter 6: The Map of s ↗