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>Introduction>What Is Euler’s Formula for Complex Numbers?

What Is Euler’s Formula for Complex Numbers?

After introducing the norm $r$ and the argument $𝜃$ , all complex numbers $z = a + b i$ can be written as

z = r (\cos 𝜃 + i \sin 𝜃) .

This expression is used to rewrite complex numbers by moving them from polar form to Cartesian form.

A complex number as a right triangle.

When you write complex numbers in this form, you can define the exponential form—often called Euler’s formula—of a complex number.

Theory

Euler’s Formula

For a complex number $z$ with norm $r$ and argument $𝜃$ the exponential form is defined as

z = r e^{i 𝜃} = r (\cos 𝜃 + i \sin 𝜃) .

In exponential form, the argument of $z$ is written in the exponent together with the imaginary unit $i$ , and the norm of $z$ is multiplied by the exponential function. Euler’s formula is an important connection between the exponential function and the trigonometric functions.

Example 1

Rewrite $z = e^{i π}$ in Cartesian form

The norm $r$ of $z$ is the number in front of the exponential function. For $z$ , you have $r = 1$ . The argument $𝜃$ of $z$ is the number standing together with $i$ in the exponent. For $z$ , you have $𝜃 = π$ . If you use Euler’s formula, you can write $z$ in Cartesian form $z = a + b i$ , where

\begin{array}{l} a & = r \cos 𝜃 = \cos π = - 1, \\ b & = r \sin 𝜃 = \sin π = 0 . \end{array}

This means that $z = e^{i π} = - 1$ .

The result from Example 1 is often called Euler’s identity, and is a known result connecting

π

i

e

and

- 1

. The exponential form is a compact way to express a complex number

z

. Euler’s formula can be used to express complex numbers in polar form. And since all complex numbers can be written in polar form, all complex numbers can also be written in exponential form.

Example 2

Rewrite $z = \sqrt{3} - i$ in polar form by using Euler’s formula

In order to use Euler’s formula, you need the norm and the argument of $z$ . You find the norm of $z$ by using the Pythagorean theorem:

\begin{array}{l} r & = \sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} \\ = \sqrt{3 + 1} \\ = 2 . \end{array}

Next, you can find the argument of $z$ by using cosine:

\begin{matrix} 𝜃 = \cos^{- 1} (\frac{\sqrt{3}}{2}) \\ ⇓ \\ 𝜃 = \frac{π}{6} or 𝜃 = \frac{11 π}{6} . \end{matrix}

Since the real part of $z$ is positive and the imaginary part of $z$ is negative, $z$ lies in the fourth quadrant of the complex plane. The argument of $z$ is therefore $𝜃 = \frac{11 π}{6}$ . Now that you have found the norm and the argument of $z$ , you can write $z$ in exponential form:

z = r e^{i 𝜃} = 2 e^{\frac{11 π}{6} i} .

When you are doing calculations with the complex exponential function, you can use normal power rules:

Rule

Complex Exponentials

For every complex number $z = a + b i$ , the exponential is

\begin{array}{l} e^{z} & = e^{a + b i} \\ = e^{a} \cdot e^{b i} \\ = e^{a} (\cos b + i \sin b) . \end{array}

e^{z} = e^{a + b i} = e^{a} \cdot e^{b i} = e^{a} (\cos b + i \sin b) .

When you raise $e$ to the power of a complex number $z = a + b i$ , you get a new complex number with norm $e^{a}$ and argument $b$ .

Example 3

Find $w = e^{z}$ for the complex number $z = 3 + i π$

You find $w$ by using the rule for complex exponentials:

\begin{array}{l} w & = e^{z} \\ = e^{3 + i π} \\ = e^{3} \cdot e^{i π} \\ = e^{3} \cdot (- 1) \\ = - e^{3} . \end{array}

w = e^{z} = e^{3 + i π} = e^{3} \cdot e^{i π} = e^{3} \cdot (- 1) = - e^{3} .

Here Euler’s identity is used to rewrite

e^{i π} = - 1

Think About This

Even though polar form and Cartesian form are equivalent ways of writing the same number

z

, the two representations have different strengths and weaknesses. Addition and subtraction of complex numbers is easiest if the numbers are written in Cartesian form. Multiplication and division of complex numbers is easiest if the numbers are written in polar form. So it’s really important to master both representations, and to be able to change between them, depending on what is most appropriate.

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What Is the Norm and the Argument of a Complex Number?

What Are Complex Conjugates?

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