FAC1004 L02 — Euler's Formula

This lecture introduces Euler's formula, which connects the exponential function with trigonometric functions in the complex plane. It covers conversion between Cartesian and exponential forms, and efficient multiplication/division of complex numbers in exponential form.

Key Concepts

  • Euler's Formula — $re^{i\theta} = r(\cos\theta + i\sin\theta)$
  • Euler's Identity — $e^{i\pi} + 1 = 0$ (special case when $r=1$, $\theta=\pi$)
  • Exponential Form — $z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$
  • Multiplication in Exponential Form — $z_1 \cdot z_2 = r_1r_2e^{i(\theta_1 + \theta_2)}$
  • Division in Exponential Form — $\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$

Euler's Formula / Identity

$$re^{i\theta} = r(\cos\theta + i\sin\theta)$$

When $r = 1$ and $\theta = \pi$:

$$e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0$$

$$e^{i\pi} + 1 = 0 \quad \text{(GOAT identity)}$$

Converting Cartesian to Exponential Form

To express $z = a + bi$ in exponential form $z = re^{i\theta}$:

  1. Compute modulus: $r = |z| = \sqrt{a^2 + b^2}$
  2. Compute argument: $\theta = \arg(z)$ (consider quadrant)

Examples

Example 1: Express $z = 1 + i$ in exponential form.

  • $r = \sqrt{1^2 + 1^2} = \sqrt{2}$
  • $\theta = \tan^{-1}(1) = \frac{\pi}{4}$ (first quadrant)
  • $z = \sqrt{2}e^{i\pi/4}$

Example 2: Write $z = 1 - \sqrt{3}i$ in exponential form.

  • $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$
  • $\theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3}$ (fourth quadrant)
  • $z = 2e^{-i\pi/3}$

Converting Exponential to Cartesian Form

Given $z = re^{i\theta}$, use Euler's formula: $$z = r(\cos\theta + i\sin\theta) = r\cos\theta + ir\sin\theta$$

So:

  • $\text{Re}(z) = r\cos\theta$
  • $\text{Im}(z) = r\sin\theta$

Examples

Example 3: Determine $\text{Re}(z)$ and $\text{Im}(z)$ for $z = 3e^{-\frac{5\pi}{6}i}$.

  • $\text{Re}(z) = 3\cos\left(-\frac{5\pi}{6}\right) = 3 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{3\sqrt{3}}{2}$
  • $\text{Im}(z) = 3\sin\left(-\frac{5\pi}{6}\right) = 3 \cdot \left(-\frac{1}{2}\right) = -\frac{3}{2}$

Finding Modulus and Argument from Exponential Form

For $z = re^{i\theta}$:

  • Modulus: $|z| = r$
  • Argument: $\arg(z) = \theta$

If the exponent has a real component, use $e^{a+bi} = e^a \cdot e^{bi}$, so:

  • $z = e^{a+bi} = e^a e^{bi} \implies |z| = e^a$, $\arg(z) = b$

Examples

Example 4: Determine modulus and argument for $z = e^{i}$.

  • $|z| = 1$, $\arg(z) = 1$ radian

Example 5: Determine modulus and argument for $z = 5e^{0.3i}$.

  • $|z| = 5$, $\arg(z) = 0.3$ radians

Example 6: Determine modulus and argument for $z = 3e^{2i}$.

  • $|z| = 3$, $\arg(z) = 2$ radians

Example 7: Determine modulus and argument for $z = e^{2+\frac{\pi}{3}i}$.

  • $z = e^2 \cdot e^{\frac{\pi}{3}i}$
  • $|z| = e^2$, $\arg(z) = \frac{\pi}{3}$

Example 8: Determine modulus and argument for $z = 3e^{2-\frac{2\pi}{3}i}$.

  • $z = 3e^2 \cdot e^{-\frac{2\pi}{3}i}$
  • $|z| = 3e^2$, $\arg(z) = -\frac{2\pi}{3}$

Multiplying and Dividing in Exponential Form

Multiplication

When multiplying complex numbers in exponential form, multiply the moduli and add the arguments:

$$z_1 \cdot z_2 = r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1 + \theta_2)}$$

Example 9: Given $z_1 = 3e^{\frac{2\pi}{3}i}$ and $z_2 = 3e^{-\frac{5\pi}{6}i}$. $$z_1 \times z_2 = 3 \cdot 3 \cdot e^{i\left(\frac{2\pi}{3} - \frac{5\pi}{6}\right)} = 9e^{i\left(\frac{4\pi - 5\pi}{6}\right)} = 9e^{-\frac{\pi}{6}i}$$

Division

When dividing complex numbers in exponential form, divide the moduli and subtract the arguments:

$$\frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$$

Example 10: Using the same $z_1, z_2$: $$z_1 \div z_2 = \frac{3}{3} \cdot e^{i\left(\frac{2\pi}{3} - \left(-\frac{5\pi}{6}\right)\right)} = e^{i\left(\frac{4\pi + 5\pi}{6}\right)} = e^{\frac{3\pi}{2}i}$$

Combined Example

Example 11: Show that $e^{1+3i} = -2.707 + 0.3836i$. $$e^{1+3i} = e^1 \cdot e^{3i} = e(\cos 3 + i\sin 3)$$ $$= e\cos 3 + ie\sin 3$$ $$\approx 2.718(-0.9900) + i(2.718)(0.1411)$$ $$\approx -2.707 + 0.3836i$$

Example 12: Let $z_1 = 2e^{-\frac{3\pi}{4}i}$ and $z_2 = \frac{3}{e^{\frac{2\pi}{3}i}} = 3e^{-\frac{2\pi}{3}i}$. Find:

  1. $\arg(z_1z_2) = -\frac{3\pi}{4} + \left(-\frac{2\pi}{3}\right) = -\frac{9\pi + 8\pi}{12} = -\frac{17\pi}{12}$
  2. $\arg\left(\frac{z_1}{z_2}\right) = -\frac{3\pi}{4} - \left(-\frac{2\pi}{3}\right) = -\frac{9\pi - 8\pi}{12} = -\frac{\pi}{12}$
  3. $|z_1 + z_2|$ — convert both to Cartesian form, add, then find modulus

Summary

Euler's formula provides a powerful connection between exponential functions and trigonometric functions in the complex plane. The exponential form $z = re^{i\theta}$ enables:

  • Easy conversion between Cartesian and polar representations
  • Efficient multiplication (multiply moduli, add arguments)
  • Efficient division (divide moduli, subtract arguments)

Related

Source File

LECTURE_NOTES_2526/L02 - Euler_s Formula Student Version .pdf