指数形式

$$\textbf{Exponential Form of a Complex Number}$$

Exponential form of a complex number looks like this: $$z=re^{i \theta}$$

We are going to derive this. Let's start by remembering how to write a complex number in polar form: $$z=r(\cos \theta + i\sin \theta)$$

Next, recall Euler's Formula: $$e^{i\theta} = \cos \theta + i\sin \theta$$

Using Euler's Formula, we can replace the $\cos \theta + i\sin \theta$ in our polar form representation with an $e^{i \theta}$ to obtain the exponential form of a complex number. $$\begin{align}z&=r(\cos \theta + i\sin \theta)\\ \Rightarrow z&=re^{i \theta} \end{align}$$

It's easy to multiply and divide complex numbers when they're in exponential form! We just need to use our exponent rules to help us.

Consider the following complex numbers in exponential form: $$\begin{align} z&=ae^{i\theta}\\ w&=be^{i\phi} \end{align}$$

$$\textbf{Multiplying Complex Numbers in Exponential Form}$$

First, let's find $zw.$ $$\begin{align} zw&=ae^{i\theta} \cdot be^{i\phi}\\ &=abe^{i\theta}e^{i\phi} \end{align}$$

Here, we can use our knowledge of exponent rules. Since $e^{i\theta}$ and $e^{i\phi}$ have the same base, we add their exponents when finding their product.

$$\begin{align} zw&=ae^{i\theta} \cdot be^{i\phi}\\ &=abe^{i\theta}e^{i\phi}\\ &=abe^{i\theta+ i\phi}, \text{ by our exponent rule for multiplication}\\ &=abe^{i(\theta+ \phi)}\\ \end{align}$$

So, for complex numbers $z=ae^{i\theta}$ and $w=be^{i\phi}$ in exponential form, their product, $zw=abe^{i(\theta+ \phi)}.$

$$\textbf{Dividing Complex Numbers in Exponential Form}$$

Next, let's find $\dfrac{z}{w}.$ $$\begin{align} \dfrac{z}{w}&=\dfrac{ae^{i\theta}}{be^{i\phi}}\\ &=\dfrac{a}{b}\dfrac{e^{i\theta}}{e^{i\phi}}\\ \end{align}$$

Here, we can use our knowledge of exponent rules. Since $e^{i\theta}$ and $e^{i\phi}$ have the same base, we use subtraction when finding their quotient.

$$\begin{align} \dfrac{z}{w}&=\dfrac{ae^{i\theta}}{be^{i\phi}}\\ &=\dfrac{a}{b}\dfrac{e^{i\theta}}{e^{i\phi}}\\ &=\dfrac{a}{b}e^{i\theta-i\phi}, \text{ by our exponent rule for division}\\ &=\dfrac{a}{b}e^{i(\theta-\phi)} \end{align}$$

So, for complex numbers $z=ae^{i\theta}$ and $w=be^{i\phi}$ in exponential form, their quotient, $\dfrac{z}{w}=\dfrac{a}{b}e^{i(\theta-\phi)}.$

Let's do an example to practice what we've learned!

First, let's find $zw.$

$$\begin{align} zw&=3e^{\frac{\pi}{2}i} \cdot 2e^{\frac{3\pi}{2}i}\\ &=(3)(2) e^{i(\frac{\pi}{2}+\frac{3\pi}{2})}, \text{ by the product formula we found before}\\ &=6e^{\frac{4\pi}{2}i}\\ &=6e^{2\pi i}\\ \end{align}$$

But remember, Euler's Formula tells us that $$e^{i\theta} = \cos \theta + i\sin \theta$$ Thus, $$\begin{align} zw=6e^{2\pi i}&=6(\cos (2\pi) + i\sin (2\pi))\\ &=6(1+i(0))\\ &=6(1)\\ &=6 \end{align}$$

Next, let's find $\dfrac{z}{w}.$ $$\begin{align} \dfrac{z}{w}&=\dfrac{3e^{\frac{\pi}{2}i}}{2e^{\frac{3\pi}{2}i}}\\ \\ &=\dfrac{3}{2}e^{i(\frac{\pi}{2}-\frac{3\pi}{2})}, \text{ by the quotient formula we found before}\\ \\ &=\dfrac{3}{2}e^{i(\frac{-2\pi}{2})}\\ \\ &=\dfrac{3}{2}e^{i(-\pi)}\\ \end{align}$$

But remember, Euler's Formula tells us that $$e^{i\theta} = \cos \theta + i\sin \theta$$ Thus, $$\begin{align} \dfrac{z}{w}=\dfrac{3}{2}e^{i(-\pi)}&=\dfrac{3}{2}(\cos (-\pi) + i\sin (-\pi))\\ &=\dfrac{3}{2}(-1 + i(0))\\ &=\dfrac{3}{2}(-1)\\ &=-\dfrac{3}{2}\\ \end{align}$$

The exponential form of a complex number is a very simple extension of its polar form. Recall that the polar form of a complex number $z$ is: $$z = r(\cos\theta+i\sin\theta) = r\mathrm{cis}\theta$$ The last expression is just a convenient shorthand for the middle expression.

Euler's Formula tells us that: $$e^{i\theta}=\cos\theta+i\sin\theta$$ Thus, we can write: $$z=re^{i\theta}$$

The exponential form of a complex number is nothing more or less than a more powerful version of the polar form. The information encapsulated in the quantities $r$ and $\theta$ is still there in almost exactly the same form. However, the exponential form allows us to make multiplication with complex numbers a piece of cake.

Just as an example, recall our formula for multiplying two complex numbers: $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta)=rs\mathrm{cis}(\alpha+\beta)$$ Well, with exponential form, watch what happens: $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta)=(re^{i\alpha})(se^{i\beta})$$ $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta)=rse^{i(\alpha+\beta)}$$ $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta)=rs\mathrm{cis}(\alpha+\beta)$$ Our seemingly incredible result now looks almost trivial in light of the even more incredible Euler's Identity.

There are many examples of the power of the exponential form, far too many to show here. Some of them may be demonstrated in the practice problems.

Exponential Form of a Complex Number

We can use the standard form for a complex number in standard form, along with Euler's Formula, to derive the exponential form of a complex number.

We know that complex numbers can be written in several different forms. The most familiar form is standard form: $$z=a+bi$$

, where $a$ is the real part, and $b$ is the imaginary part.

However, complex numbers can also be written in polar form and exponential form. These forms are especially useful when multiplying, dividing, and raising complex numbers to powers.

Suppose a complex number is: $z=a+bi$

We can represent this number as a point, $(a,b)$, on the complex plane. To write the number in polar form, we need two quantities.

The first is the modulus. The modulus, denoted by $r$, is the distance from the origin to the point.

$$r=\sqrt{a^2+b^2}$$

The second is the argument. The argument, denoted by $\theta$, is the angle formed with the positive real axis. Here is the formula to find the argument:

$$\tan\theta = \frac{b}{a}$$

After determining the correct quadrant, the complex number can be written as:

$$z=r(\cos\theta + i\sin\theta)$$

This is called the polar form of a complex number.

####Euler's Formula

The connection between polar form and exponential form comes from one of the most important formulas in mathematics:

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

This relationship is known as Euler's Formula.

Because $\cos\theta + i\sin\theta = e^{i\theta}$, the polar form can be written more simply.

####Exponential Form

Substituting Euler's Formula into polar form gives us:

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

This is the exponential form of a complex number. It contains the same information as the polar form, but it tends to be easier to work with.

Looking Further

In the next lesson, you will learn about multiplying and dividing complex numbers in polar form. We're continuing to build on this exciting topic!

(Video) Complex Numbers - Exponential Form

by ExamSolutions

In this video by ExamSolutions, you will learn how to express a complex number in exponential form.