Euler's Formula — Physica

Equivalent forms

e^{i\pi} + 1 = 0

\cos\theta = \tfrac{1}{2}(e^{i\theta}+e^{-i\theta}),\quad \sin\theta = \tfrac{1}{2i}(e^{i\theta}-e^{-i\theta})

z = re^{i\theta}

Five fundamental constants — 0, 1, e, i, π — bound into one identity, and every rotation, oscillation, and AC circuit secretly runs on it.

Symbol	Name	SI unit	Dimension	Range
$\theta$	Angle Phase angle / argument	rad	1	-6.283 – 6.283
$\cos\theta$	Real partoutput Horizontal projection of e^{iθ}	1	1	-1 – 1
$\sin\theta$	Imaginary partoutput Vertical projection of e^{iθ}	1	1	-1 – 1

Unit systems

Symbol	SI	CGS	Imperial
$\theta$	rad	rad	rad
$\cos\theta$	1	1	1
$\sin\theta$	1	1	1

Where it holds

All real

\theta

; extends to all complex z via

e^z = e^x(\cos y + i \sin y)

.

Dimensional analysis

e^{i\theta }

→ dimensionless

\cos \theta + i \sin \theta

→ dimensionless

Arguments of exp/sin/cos must be dimensionless;

\theta in

radians is a pure number, and the result is a pure complex number of modulus 1.

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Discovery

Leonhard Euler · 1748

Euler published the formula in his 1748 Introductio, uniting the exponential, trigonometric, and imaginary worlds. The special case e^{iπ}+1=0 — linking e, i, π, 1 and 0 — is routinely voted the most beautiful equation in mathematics.

Try this

How can raising a number to an imaginary power make it spin? Euler's formula says exponential growth and circular motion are the same act, seen from different sides.

Spin the angle θ and watch e^{iθ} ride the unit circle while its shadow traces cosθ and its height traces sinθ.

Research status: stable

Real-world applications

Phasors in AC circuit analysis and signal processing
Fourier analysis $(e^{i\omega t}$ basis functions)
Quantum mechanics wavefunctions and time evolution $e^{-iEt/\hbar }$
Rotations in computer graphics and control theory

Common misconceptions

$e^{i\theta }$ grows like a normal exponential — it has constant magnitude 1; only its phase changes.
i is 'imaginary' so it isn't physical — it is just a $90^{\circ}$ rotation operator and is everywhere in engineering.
Euler's identity is a coincidence — it's a direct consequence of the series definitions.

Experimental verification

Underlies all phasor analysis in AC electronics — every oscilloscope trace of voltage and current phase agrees with

e^{i\omega t}

predictions — and the interference fringes of optics and quantum mechanics.

Derivation

Compare Taylor series:

e^{i\theta } = \Sigma (i\theta )^{n}/n!

.

Splitting even and odd powers, the even terms (real) sum to

1 - \theta ^{2}/2! + \theta ^{4}/4! -

…

= \cos \theta

, and the odd terms (imaginary) sum to

i(\theta - \theta ^{3}/3!

+ …

) = i \sin \theta

.

Hence

e^{i\theta } = \cos \theta + i \sin \theta

.

Limiting cases

\theta = 0

⟶

e^{i_{0}} = 1

No rotation; the phasor sits on the positive real axis.

\theta = \pi

⟶

e^{i\pi } = -1

Half turn lands

on -1

— Euler's identity.

\theta = \pi /2

⟶

e^{i\pi /2} = i

Quarter turn maps 1 to i — multiplication by i is a

90^{\circ}

rotation.

What if…

What if the exponent is complex, a + bi?

e^{a+ $bi} = e$ ^a(cos b + i sin b): the real part scales the magnitude, the imaginary part rotates the phase.

What if I average

e^{i\theta }

and

e^{-i\theta }

?

You get $\cos \theta$ — counter-rotating phasors add to a real oscillation, the heart of Fourier analysis.

1

Evaluate e^{iπ/3}

Given ·

\theta:: $\pi /3$

Find · rectangular form

Steps

$\cos (\pi /3) = 1/2$
$\sin (\pi /3) = \sqrt{3}/2 \approx 0.866$
$e^{i\pi /3} = 0.5 + 0.866\,\mathrm{i}$

Result ·

e^{i\pi /3} = \cos 60^{\circ} + i \sin 60^{\circ} = 1/2 + i\sqrt{3}/2

2

Why i·i = −1, geometrically

Given ·

i:: $e^{i\pi /2}$

Find ·

i^{2}

Steps

Each i is a $90^{\circ}$ rotation
Two of them make a $180^{\circ}$ rotation
$180^{\circ}$ rotation of $1\,\mathrm{is} -1$

Result ·

i^{2} = e^{i\pi /2}\cdot e^{i\pi /2} = e^{i\pi } = -1

de moivre theoremFourier Series→Cauchy–Riemann Equations→taylor series complex