The Exponential Form of a Complex Number

The exponential series, fed an imaginary number

The Maclaurin series for $e^{x}$ is $e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots$, and it adds up to $e^{x}$ for every real number x. Its terms use only multiplying and adding, which work for complex numbers too, so the same series gives a meaning to e raised to an imaginary power.

Substitute $x = i\theta$, where $\theta$ is real: $e^{i\theta} = 1 + i\theta + \frac{\left(i\theta\right)^{2}}{2!} + \frac{\left(i\theta\right)^{3}}{3!} + \frac{\left(i\theta\right)^{4}}{4!} + \dots$

The powers of i sort the terms

The powers of i repeat every four: $i^{0} = 1$, $i^{1} = i$, $i^{2} = -1$, $i^{3} = -i$, and then $i^{4} = 1$ again. So $\left(i\theta\right)^{2} = -\theta^{2}$, $\left(i\theta\right)^{3} = -i\theta^{3}$, $\left(i\theta\right)^{4} = \theta^{4}$ and $\left(i\theta\right)^{5} = i\theta^{5}$.

The series becomes $1 + i\theta - \frac{\theta^{2}}{2!} - \frac{i\theta^{3}}{3!} + \frac{\theta^{4}}{4!} + \frac{i\theta^{5}}{5!} - \dots$ The even powers have no i and the odd powers have one, so collect them separately: $e^{i\theta} = \left(1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} - \dots\right) + i\left(\theta - \frac{\theta^{3}}{3!} + \frac{\theta^{5}}{5!} - \dots\right)$.

Cosine and sine

The bracket without i is the Maclaurin series for $\cos \theta$, and the bracket with i is the series for $\sin \theta$. So $e^{i\theta} = \cos \theta + i \sin \theta$.

From here on the angle is in RADIANS. The series $\cos \theta = 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} - \dots$ and $\sin \theta = \theta - \frac{\theta^{3}}{3!} + \dots$ hold only for $\theta$ in radians, so $e^{i\theta} = \cos \theta + i \sin \theta$ does too. A half turn is $\pi$ radians, so 30° is $\frac{\pi}{6}$, 60° is $\frac{\pi}{3}$, 90° is $\frac{\pi}{2}$ and 180° is $\pi$.

Check at $\theta = 1$, which is 1 radian, about 57.3°. The first four terms of each bracket give $1 - \frac{1}{2} + \frac{1}{24} - \frac{1}{720} \approx 0.5403$ and $1 - \frac{1}{6} + \frac{1}{120} - \frac{1}{5040} \approx 0.8415$, and $\cos 1 \approx 0.5403$, $\sin 1 \approx 0.8415$.

The terms of the series for $e^{i\theta}$ at $\theta = 1$, laid head to tail: 1, then i, then $- \frac{1}{2}$, then $- \frac{i}{6}$, then $\frac{1}{24}$. Each turns a quarter turn from the one before, and from the third on each is shorter; their total closes on the point $\cos 1 + i \sin 1 \approx 0.540 + 0.841i$, on the dashed unit circle.

A point on the unit circle

The length of $e^{i\theta}$ is $\sqrt{\cos^{2}\theta + \sin^{2}\theta} = 1$, and its argument is $\theta$. So $e^{i\theta}$ is the point on the unit circle at angle $\theta$ from the positive real axis.

The quarter turns: $e^{i\cdot0} = 1$, $e^{\frac{i\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) = i$, $e^{i\pi} = \cos \pi + i \sin \pi = -1$, and $e^{- \frac{i\pi}{2}} = -i$. Written another way, $e^{i\pi} + 1 = 0$. A full turn returns to the start: $e^{2\pi i} = 1$.

$e^{\frac{i\pi}{3}}$ is the point at 60°: $\cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) = \frac{1}{2} + \left(\frac{\sqrt{3}}{2}\right)i \approx 0.5 + 0.866i$.

Five values of $e^{i\theta}$ on the unit circle: 1 at $\theta = 0$, $\frac{1}{2} + \left(\frac{\sqrt{3}}{2}\right)i$ at $\frac{\pi}{3}$, i at $\frac{\pi}{2}$, −1 at $\pi$ and −i at $- \frac{\pi}{2}$.

Every complex number as re$^{i\theta}$

A complex number with modulus r and argument $\theta$ is $z = r\left(\cos \theta + i \sin \theta\right)$, and the bracket is $e^{i\theta}$. So z = re$^{i\theta}$: r is the distance from the origin, and $\theta$, in radians, is the direction.

$2\left(\cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right)\right) = 2e^{\frac{i\pi}{3}}$, which is $1 + \sqrt{3} i$. 1 + i has modulus $\sqrt{2}$ and argument $\frac{\pi}{4}$, so $1 + i = \sqrt{2} e^{\frac{i\pi}{4}}$. $1 - i = \sqrt{2} e^{- \frac{i\pi}{4}}$, its argument below the axis.

On the axes: $5 = 5e^{i\cdot0}$, $3i = 3e^{\frac{i\pi}{2}}$, $-3 = 3e^{i\pi}$ and $-2i = 2e^{- \frac{i\pi}{2}}$. Arguments are taken in $-\pi < \theta \le \pi$, the radian form of $-180^\circ < \theta \le 180^\circ$.

Lengths multiply, angles add

Index laws add powers: $e^{a} \times e^{b} = e^{a + b}$. So $r_{1}e^{i\theta_{1}} \times r_{2}e^{i\theta_{2}} = r_{1}r_{2} e^{i\left(\theta_{1} + \theta_{2}\right)}$. The lengths multiply and the angles add, the rule for multiplying complex numbers met in polar form.

$2e^{\frac{i\pi}{6}} \times 3e^{\frac{i\pi}{3}} = 6e^{i\left(\frac{\pi}{6} + \frac{\pi}{3}\right)} = 6e^{\frac{i\pi}{2}} = 6i$. In parts, $2e^{\frac{i\pi}{6}} = \sqrt{3} + i$ and $3e^{\frac{i\pi}{3}} = \frac{3}{2} + \left(\frac{3\sqrt{3}}{2}\right)i$, and $\left(\sqrt{3} + i\right)\left(\frac{3}{2} + \left(\frac{3\sqrt{3}}{2}\right)i\right) = \frac{3\sqrt{3}}{2} + \left(\frac{9}{2}\right)i + \left(\frac{3}{2}\right)i + \left(\frac{3\sqrt{3}}{2}\right)i^{2} = 0 + 6i$.

Dividing subtracts the angles: $6e^{\frac{i\pi}{2}} / 2e^{\frac{i\pi}{6}} = 3e^{\frac{i\pi}{3}}$. The conjugate of re$^{i\theta}$ is re$^{-i\theta}$. And De Moivre’s theorem is the index law $\left(e^{i\theta}\right)^{n} = e^{(}$in$\theta)$.

The usual mistakes

Moving r into the exponent. $2e^{\frac{i\pi}{3}}$ is not $e^{\frac{2i\pi}{3}}$: the second has length 1 and angle $\frac{2\pi}{3}$.

Dropping the i. $2e^{\frac{\pi}{3}}$ is a real number, about 5.70; it is the i in the exponent that makes an angle.

Giving $e^{i\pi} = 1$. $\pi$ is half a turn, so $e^{i\pi} = -1$; a full turn, $e^{2\pi i}$, is 1.

Adding the lengths, or multiplying the exponents. $2e^{\frac{i\pi}{6}} \times 3e^{\frac{i\pi}{3}}$ is $6e^{\frac{i\pi}{2}}$, not $5e^{\frac{i\pi}{2}}$ or $6e^{\frac{i\pi^{2}}{18}}$.

Putting degrees in the exponent. $e^{60i}$ is not $e^{\frac{i\pi}{3}}$: it is the point at 60 radians, about −0.952 − 0.305i.

Four loudspeakers slightly out of step

In the application below, the phasors 1, $e^{\frac{i\pi}{6}}$, $e^{\frac{2i\pi}{6}}$ and $e^{\frac{3i\pi}{6}}$ are 1 at 0°, 30°, 60° and 90°. Their sum is the sum drawn in Summing a Series with De Moivre at $\theta = 30^\circ$: $\frac{{3 + \sqrt{3}}}{2} + \left(\frac{{3 + \sqrt{3}}}{2}\right)i \approx 2.366 + 2.366i$, which has length about 3.35 and argument $\frac{\pi}{4}$, or 45°.