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# 5. Exponential Form of a Complex Number

by M. Bourne

IMPORTANT:

In this section, θ MUST be expressed in radians.

We use the important constant

e = 2.718 281 8...

in this section.

We first met e in the section Natural logarithms (to the base e).

The exponential form of a complex number is:

r e^(\ j\ theta)

(r is the absolute value of the complex number, the same as we had before in the Polar Form;
j=sqrt(-1).

### Example 1

Express 5(cos 135^@ +j\ sin\ 135^@) in exponential form.

We have r = 5 from the question.

We must express  θ = 135^@ in radians.

Recall:

1^text(o)=pi/180

So

135^text(o)=(135pi)/180

=(3pi)/4

~~2.36 radians

So we can write

5(cos\ 135^text(o)+j\ sin135^text(o))

=5e^((3pij)/4)

 ~~ 5e^(2.36j)

### Example 2

Express -1 + 5j in exponential form.

We need to find θ in radians (see Trigonometric Functions of Any Angle if you need a reminder about reference angles) and r.

alpha=tan^(-1)(y/x) =tan^(-1)(5/1) ~~1.37text( radians)

[This is 78.7^@ if we were working in degrees.]

Because our angle is in the second quadrant, we need to apply:

theta = pi - 1.37 ~~1.77

And

r=sqrt(x^2+y^2)

=sqrt( (-1)^2 + (5)^2 )

= sqrt(26)

 ~~ 5.10

So -1 + 5j in exponential form is 5.10e^(1.77j)

## SUMMARY: Forms of a complex number

These expressions have the same value. They are just different ways of expressing the same complex number.

x + yj

### b. Polar form

r(cos θ + j sin θ) = r cis θ = r∠θ

θ can be in degrees OR radians for Polar form.

### c. Exponential form

rejθ

θ MUST be in radians for Exponential form.

## Exercises

1. Express in exponential form:

4.50(cos\ 282.3^@+ j\ sin\ 282.3^@)

282.3^@ = 4.93 radians

So

4.50(cos\ 282.3^@ + j\ sin\ 282.3^@)  = 4.50e^(4.93j)

2. Express in exponential form: -1 - 5j

This is similar to our -1 + 5j example above, but this time we are in the 3rd quadrant.

r=sqrt(1+25) =sqrt(26)~~5.10

θ = π + 1.37 = 4.51 radians

The graph for this example:

So

-1-5j=5.10\ e^(4.51j)

3. Express in polar and rectangular forms: 2.50e^(3.84j)

We can immediately write:

2.50e^(3.84j) = 2.50\ /_ \ 3.84 [polar form, θ in radians]

OR, if you prefer, since 3.84\ "radians" = 220^@,

2.50e^(3.84j)  = 2.50(cos\ 220^@ + j\ sin\ 220^@) [polar form, θ in degrees]

And, using this result, we can multiply the right hand side to give:

2.50(cos\ 220^@ + j\ sin\ 220^@)  = -1.92 -1.61j

Summary

Our complex number can be written in the following equivalent forms:

2.50e^(3.84j) [exponential form]

 2.50\ /_ \ 3.84 =2.50(cos\ 220^@ + j\ sin\ 220^@) [polar form]

-1.92 -1.61j [rectangular form]

### Euler's Formula and Identity

The next section has an interactive graph where you can explore a special case of Complex Numbers in Exponential Form:

Euler Formula and Euler Identity interactive graph

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