1.5 Exponential form of complex numbers

The exponential form of complex numbers bridges algebra and geometry, offering a powerful tool for manipulating these mathematical objects. By expressing complex numbers as $re^{i\theta}$, we can easily multiply, divide, and find powers and roots. This representation simplifies calculations and provides insight into the geometric properties of complex numbers.

Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, is the foundation of the exponential form. It connects exponential and trigonometric functions, allowing us to switch between rectangular and polar forms effortlessly. This versatility makes the exponential form invaluable for solving complex equations and understanding their geometric interpretations.

Complex numbers in exponential form

Euler's formula and the exponential form

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• Euler's formula states that for any real number x, $e^{ix} = \cos(x) + i\sin(x)$
  • This formula connects the exponential function with trigonometric functions
• The exponential form of a complex number $z$ is $z = re^{i\theta}$, where $r$ is the modulus (magnitude) and $\theta$ is the argument (angle) in radians
  • In the complex plane, the modulus $r$ represents the distance from the origin to the point representing the complex number
  • The argument $\theta$ is the angle formed with the positive real axis
• The real part of $z$ is $r\cos(\theta)$ and the imaginary part is $r\sin(\theta)$
• The exponential form allows for easier manipulation of complex numbers when multiplying, dividing, or finding powers and roots

Applications and advantages of the exponential form

• The exponential form simplifies calculations involving powers and roots of complex numbers by utilizing the properties of exponents and trigonometric functions
• When multiplying complex numbers in exponential form, the moduli are multiplied and the arguments are added: $z_1z_2 = r_1r_2e^{i(\theta_1+\theta_2)}$
• When dividing complex numbers in exponential form, the moduli are divided and the arguments are subtracted: $\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$
• To raise a complex number $z = re^{i\theta}$ to a power $n$, use the formula: $z^n = (re^{i\theta})^n = r^n e^{in\theta}$
• When finding the $n$th roots of a complex number $z$, consider the formula: $z^{\frac{1}{n}} = r^{\frac{1}{n}} e^{i(\frac{\theta + 2k\pi}{n})}$, where $k = 0, 1, \ldots, n-1$

Exponential, polar, and rectangular forms

Converting between different forms of complex numbers

• The polar form of a complex number is $z = r\text{cis}(\theta)$, where $\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)$
  • This is equivalent to the exponential form $z = re^{i\theta}$
• To convert from rectangular form $(a + bi)$ to polar or exponential form:
  • Calculate the modulus using $r = \sqrt{a^2 + b^2}$
  • Calculate the argument using $\theta = \text{atan2}(b, a)$, where $\text{atan2}$ is the two-argument arctangent function
• To convert from polar or exponential form to rectangular form, use the formulas:
  • Real part: $a = r\cos(\theta)$
  • Imaginary part: $b = r\sin(\theta)$
• Example: Convert $2 + 2i$ to polar form
  • $r = \sqrt{2^2 + 2^2} = 2\sqrt{2}$
  • $\theta = \text{atan2}(2, 2) = \frac{\pi}{4}$
  • Polar form: $z = 2\sqrt{2}\text{cis}(\frac{\pi}{4})$ or $z = 2\sqrt{2}e^{i\frac{\pi}{4}}$

Choosing the appropriate form for a given problem

• When working with complex numbers in different forms, it's essential to consider the context and choose the most appropriate representation for the given problem
• Rectangular form $(a + bi)$ is often used when performing addition, subtraction, or when the real and imaginary parts are of interest
• Polar form $r\text{cis}(\theta)$ and exponential form $re^{i\theta}$ are useful when multiplying, dividing, or finding powers and roots of complex numbers
• Example: Solve the equation $z^4 = 16i$
  • Converting to exponential form: $z = 2e^{i\frac{\pi}{2}}$
  • Using De Moivre's formula: $z^4 = (2e^{i\frac{\pi}{2}})^4 = 16e^{i2\pi} = 16$
  • The exponential form simplifies the calculation of the fourth power

Powers and roots of complex numbers

Applying the exponential form to solve problems

• To raise a complex number $z = re^{i\theta}$ to a power $n$, use the formula: $z^n = (re^{i\theta})^n = r^n e^{in\theta}$
• When finding the $n$th roots of a complex number $z$, consider the formula: $z^{\frac{1}{n}} = r^{\frac{1}{n}} e^{i(\frac{\theta + 2k\pi}{n})}$, where $k = 0, 1, \ldots, n-1$
• The $n$th roots of unity are complex numbers that satisfy the equation $z^n = 1$
  • They are evenly spaced points on the unit circle in the complex plane and can be expressed as $e^{i\frac{2k\pi}{n}}$, where $k = 0, 1, \ldots, n-1$
• Example: Find the cube roots of $-8$
  • In exponential form: $-8 = 8e^{i\pi}$
  • Using the formula: $z^{\frac{1}{3}} = 2e^{i(\frac{\pi + 2k\pi}{3})}$, where $k = 0, 1, 2$
  • The three cube roots are: $2e^{i\frac{\pi}{3}}, 2e^{i\pi}, 2e^{i\frac{5\pi}{3}}$

Solving equations and simplifying expressions

• The exponential form simplifies calculations involving powers and roots of complex numbers by utilizing the properties of exponents and trigonometric functions
• When solving equations or simplifying expressions involving complex numbers, converting to exponential form can often lead to more straightforward solutions
• Example: Simplify $(1 + i)^6$
  • Converting to exponential form: $1 + i = \sqrt{2}e^{i\frac{\pi}{4}}$
  • Using De Moivre's formula: $(1 + i)^6 = (\sqrt{2}e^{i\frac{\pi}{4}})^6 = 8e^{i\frac{3\pi}{2}} = -8i$
  • The exponential form allows for the application of De Moivre's formula, simplifying the calculation

De Moivre's formula for complex numbers

Derivation and application of De Moivre's formula

• De Moivre's formula is a generalization of Euler's formula for integer powers of complex numbers
  • It states that for any complex number $z$ and any integer $n$: $(\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)$
• To derive De Moivre's formula, start with Euler's formula and apply the binomial theorem to expand $(e^{i\theta})^n$
• De Moivre's formula is particularly useful for finding powers and roots of complex numbers expressed in polar or exponential form
• When using De Moivre's formula to find the $n$th roots of a complex number $z$, the solutions are given by: $z^{\frac{1}{n}} = r^{\frac{1}{n}} (\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$, where $k = 0, 1, \ldots, n-1$
• Example: Find the fourth roots of $16(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))$
  • Using De Moivre's formula: $z^{\frac{1}{4}} = 2(\cos(\frac{\frac{\pi}{3} + 2k\pi}{4}) + i\sin(\frac{\frac{\pi}{3} + 2k\pi}{4}))$, where $k = 0, 1, 2, 3$
  • The four roots are: $2(\cos(\frac{\pi}{12}) + i\sin(\frac{\pi}{12})), 2(\cos(\frac{7\pi}{12}) + i\sin(\frac{7\pi}{12})), 2(\cos(\frac{13\pi}{12}) + i\sin(\frac{13\pi}{12})), 2(\cos(\frac{19\pi}{12}) + i\sin(\frac{19\pi}{12}))$

Solving trigonometric identities and equations

• Applying De Moivre's formula can simplify complex number calculations and help solve problems involving trigonometric identities and equations
• By expressing trigonometric functions in terms of complex exponentials using Euler's formula, De Moivre's formula can be applied to simplify expressions and solve equations
• Example: Prove the identity $\cos(5\theta) = 16\cos^5(\theta) - 20\cos^3(\theta) + 5\cos(\theta)$
  • Using Euler's formula: $\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$
  • Applying De Moivre's formula: $(\frac{e^{i\theta} + e^{-i\theta}}{2})^5 = \frac{e^{i5\theta} + e^{-i5\theta}}{2} = \cos(5\theta)$
  • Expanding the left side using the binomial theorem and simplifying leads to the desired identity
• Mastering the use of De Moivre's formula is crucial for solving advanced problems involving complex numbers and trigonometric functions