concept

Parameterization and substitution

Formulas are reused by substitution: an expression takes the place of a variable, as when θ\theta becomes 2πft2\pi f t, or xx becomes iθi\theta. Those two examples are different moves with different justifications. This page separates them: composition, which supplies new inputs to an unchanged function, and extension, which enlarges the set of inputs the function accepts. Identifying which move was made determines exactly what remains true afterward.

The second widget uses complex numbers and relates to Euler's formula.

Parameters fill slots

A formula like Asin⁡θA \sin\theta has slots. AA is a slot for a size; θ\theta is a slot for an angle in radians. Parameterizing a slot means filling it with a function of some other quantity, so that the formula's output follows that quantity. The angle slot of every sinusoid in these courses is filled the same way:

θ(t)=2πf t\theta(t) = 2\pi f\, t

tt is time in seconds and ff is frequency in rotations per second. The product f tf\,t counts rotations completed by time tt, and the factor 2π2\pi converts rotations to radians, the unit the angle slot expects. The frequency ff is the parameter: each choice of ff gives a different schedule for the same angle variable. Nothing about sin⁡\sin changes; only the source of its input does.

Move one: composition

Writing sin⁡(2πft)\sin(2\pi f t) is the composition sin⁡(θ(t))\sin(\theta(t)): the unchanged sine function, evaluated along a moving input. Any expression may fill a slot, provided it produces a value of the kind the slot expects; here that kind is an angle in radians. Because the function itself is untouched, every property of sin⁡\sin still holds at each instant; the only new fact is the input's schedule. Rates compose by the chain rule: the angle advances at dθ/dt=2πfd\theta/dt = 2\pi f radians per second, which is ff rotations per second.

The time bar on the left is the input tt. The circle on the right shows the angle θ=2πft\theta = 2\pi f t that the substitution produces, with the swept arc shaded and full rotations counted. Doubling ff doubles the rate at which θ\theta accumulates and changes nothing else: the circle, the sine, and the radian are all unchanged.

Move two: extension

Writing eiθe^{i\theta} is not composition, because the original exe^{x} is defined for real xx and iθi\theta is not a real number. The substitution is justified differently. The exponential is defined by a construction:

ez  :=  lim⁡n→∞(1+zn) ⁣ne^{z} \;:=\; \lim_{n \to \infty}\Big(1 + \frac{z}{n}\Big)^{\!n}

zz is the input, and the limit is taken over the positive integers nn. The construction uses only addition and multiplication. Complex numbers have both operations, so the construction accepts complex inputs unchanged, and its output for complex zz defines the extended function. The old function is never evaluated outside its domain; a new function is built from the same construction, and the new function agrees with the old one wherever both are defined, that is, for real zz.

What survives the extension is precisely what can be proved from the construction using only addition and multiplication. The multiplication law survives: ez+w=ezewe^{z+w} = e^{z} e^{w} holds for any complex zz and ww. The split into growth and rotation also survives, and it is the structure the courses rely on:

∣e a+bi∣=ea,∠ e a+bi=b\big|e^{\,a + bi}\big| = e^{a}, \qquad \angle\, e^{\,a + bi} = b

In the input a+bia + bi, the real part aa controls only the length of the output, and the imaginary part bb controls only its angle. Properties that mention order do not survive: “exe^{x} is positive” and “exe^{x} is increasing” are statements about the real number line, and complex numbers have no such order.

The widget tests the surviving laws. Drag zz anywhere and compare the measured length and angle of the construction's output against the predictions eae^{a} and bb. Place zz on the imaginary axis, where a=0a = 0, and the predicted length stays at 1; place it on the real axis, where b=0b = 0, and the predicted angle stays at 0.

The left plane shows the input zz, which can be dragged. The right plane shows the partial products of (1+z/n)n(1 + z/n)^{n} spiraling toward eze^{z}. A purely imaginary zz produces an output on the unit circle, which is Euler's formula as the special case a=0a = 0. A purely real zz produces an output on the real axis, with no rotation. A mixed zz produces both effects at once, and the partial products trace a spiral. The two readouts agree at every position, and that agreement is the surviving structure of the extension.

The chain, assembled

The sequence from the courses uses both moves, one after the other:

ex  →  x  =  iθ    eiθ  →  θ  =  2πft    ei2πft  →  θ  →  θ+φ    ei(2πft+φ)e^{x} \;\xrightarrow{\;x \;=\; i\theta\;}\; e^{i\theta} \;\xrightarrow{\;\theta \;=\; 2\pi f t\;}\; e^{i 2\pi f t} \;\xrightarrow{\;\theta \;\to\; \theta + \varphi\;}\; e^{i(2\pi f t + \varphi)}

The first arrow is the extension (move two): a one-time enlargement of the function's domain, justified by the construction. Every arrow after it is plain composition (move one): θ\theta is an angle slot, 2πft2\pi f t is an angle-valued function of time parameterized by frequency, and adding φ\varphi shifts the starting angle. After the first arrow, the remaining substitutions are the same move as writing sin⁡(2πft)\sin(2\pi f t).

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In the web

Builds on the assumed floor only — a foundation.