Euler's formula

Euler's formula

Lesson 1.4.2 built $e^x$ as the limit of $\left(1 + \frac{x}{n}\right)^n$: split a growth $x$ into $n$ small pieces. Now try an imaginary growth, $x = i\theta$, where $\theta$ is an angle in radians. Each factor $1 + \frac{i\theta}{n}$ is a complex number, so multiplying by it turns a point as well as stretching it.

Take $\theta = \pi$. The table shows $\left(1 + \frac{i\pi}{n}\right)^n$ as $n$ grows.

The values close in on $-1$. Multiplying a point $z$ by $1 + \frac{i\pi}{n}$ adds $\frac{i\pi}{n}z$ to it. Since multiplying by $i$ is a quarter turn (lesson 1.7.2), that is a short step at right angles to $z$.

Steps at right angles to the radius go round a circle. Their lengths add up to about $\pi$, so the path covers about $\pi$ of the unit circle. That is half a turn, from 1 to $-1$.

The same happens for any angle: the limit is the point an arc $\theta$ round the unit circle from 1. Lesson 1.7.1 named that point $(\cos\theta, \sin\theta)$. Written as a complex number, this is Euler's formula.

Why this holds for every $\theta$ needs calculus; module 2.7 (Taylor series) shows one way to see it. Here the numbers are the evidence, as they were for $e$ in lesson 1.4.2.

Going round the unit circle

Every value of $e^{i\theta}$ lies on the unit circle. Its distance from 0 is $\sqrt{\cos^2\theta + \sin^2\theta} = 1$, by the identity of lesson 1.7.1. The quarter turns give four values worth knowing.

So $e^{i\pi} = -1$: two famous numbers, $e$ and $\pi$, combine to give $-1$. A full turn, $e^{2\pi i}$, is back at 1.

The facts about angles from lesson 1.7.1 carry over:

• Period $2\pi$: $e^{i(\theta + 2\pi)} = e^{i\theta}$, because one more full turn changes nothing.
• Negative angles: $e^{-i\theta} = \cos\theta - i\sin\theta$, the conjugate of $e^{i\theta}$. Turning clockwise mirrors the point in the real axis.
• Degrees do not work: $e^{90i}$ is the point 90 radians round, about $-0.448074 + 0.893997i$, not $i$.

Example 1. Write $e^{i\pi/6}$ and $e^{-i\pi/6}$ in the form $a + bi$. By the exact values of lesson 1.7.1, $e^{i\pi/6} = \frac{\sqrt{3}}{2} + \frac{1}{2}i \approx 0.866025 + 0.5i$. Its conjugate is $e^{-i\pi/6} \approx 0.866025 - 0.5i$.

Polar form

Any complex number other than 0 is a distance from 0 and a direction. Let $r = |z|$ be the distance and $\theta$ the angle from the positive real axis to $z$, called its argument. Then $z$ is $r$ times the point at angle $\theta$ on the unit circle.

From polar form to $a + bi$, use the cosine and sine. For example $2e^{i\pi/3} = 2\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = 1 + \sqrt{3}\,i \approx 1 + 1.732051i$.

From $a + bi$ to polar form, $r$ is the modulus and $\theta$ the angle of the point $(a, b)$. For $1 + i$, $r = \sqrt{2} \approx 1.414214$ and $\theta = \frac{\pi}{4}$, which is 45°.

The quadrant matters. $-1 - i$ also has $r = \sqrt{2}$, and the same slope $\frac{b}{a} = 1$ as $1 + i$. But it lies in quadrant III, so $\theta = -\frac{3\pi}{4}$ ($-135°$), not $\frac{\pi}{4}$.

Four more cases come up often:

• Angles are not unique. Adding $2\pi$ gives the same number, so $-\frac{3\pi}{4}$ and $\frac{5\pi}{4}$ both describe $-1 - i$. Python's cmath.phase picks the angle between $-\pi$ and $\pi$.
• Real numbers: a positive one has angle 0, and a negative one has angle $\pi$. So $-2 = 2e^{i\pi}$.
• A negative $r$ is not polar form. $-3e^{i\pi/4}$ points the opposite way, so it is $3e^{i(\pi/4 + \pi)} = 3e^{5i\pi/4}$. Python reports its angle as $-\frac{3\pi}{4} \approx -2.356194$.
• 0 has no angle. Its distance is 0, and every angle gives the same point.

Multiplying: a stretch and a turn

In polar form, multiplication needs only the exponent rules of lesson 1.4.1. Write $\alpha$ and $\beta$ (the Greek letters alpha and beta) for two angles.

So multiplying by $e^{i\beta}$, at distance 1, turns a point through $\beta$ without stretching it. Multiplying by a positive real number $r$ stretches it without turning. Since $i = e^{i\pi/2}$, multiplying by $i$ is the quarter turn of lesson 1.7.2.

Example 2. Multiply $z = 1 + i$ by $w = 1 + \sqrt{3}\,i$.

In polar form, $z$ is $\sqrt{2}$ at 45° and $w$ is 2 at 60°. So $zw$ is $2\sqrt{2} \approx 2.828427$ at 105°. Multiplying out instead (lesson 1.7.2) gives the same number:

The conjugate fits the same picture. $\bar{z} = re^{-i\theta}$ has the opposite angle, so $z\bar{z} = r^2 e^{0} = r^2$: the $|z|^2$ of lesson 1.7.2. Division works the same way: $6e^{5i\pi/6} \div 2e^{i\pi/3} = 3e^{i\pi/2} = 3i$.

Powers

The power $z^n$ of $z = re^{i\theta}$ multiplies $n$ copies of $z$. Each copy multiplies the distance by $r$ and adds $\theta$ to the angle.

Example 3. Find $(1 + i)^8$. Each power multiplies the distance by $\sqrt{2}$ and adds 45°, so the points spiral outwards.

The power rule gives the last entry at once: $\left(\sqrt{2}\,e^{i\pi/4}\right)^8 = (\sqrt{2})^8 e^{8i\pi/4} = 16\,e^{2\pi i} = 16$.

Module 5.6 builds characteristic functions of random variables from $e^{i\theta}$, and the Financial Computing with Python course prices options from them by Fourier methods.

Common mistakes

1. The angle must be in radians. $e^{90i}$ is about $-0.448074 + 0.893997i$, not $i$. A quarter turn is $e^{i\pi/2}$.
2. $|e^{i\theta}| = 1$, not $e^{\theta}$. An imaginary exponent turns the point; it does not grow it. $|e^{3i}| = 1$, while $e^3 \approx 20.085537$.
3. Check the quadrant. $-1 - i$ and $1 + i$ share the slope $\frac{b}{a} = 1$, but their angles are $-135°$ and 45°.
4. Angles add only when you multiply. $1$ is at 0° and $i$ at 90°, but their sum $1 + i$ is at 45°. Its distance is $\sqrt{2}$, not 2.
5. The distance $r$ is never negative. Rewrite $-3e^{i\pi/4}$ as $3e^{i(\pi/4 + \pi)}$.
6. An angle has many names. $\frac{3\pi}{2}$ and $-\frac{\pi}{2}$ are the same direction; Python reports the one between $-\pi$ and $\pi$.