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5. Exponential Form of a Complex Number

by M. Bourne


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;
θ is in radians; and

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.






`~~2.36` radians

So we can write

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


` ~~ 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`



`=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.

a. Rectangular form

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


θ MUST be in radians for Exponential form.


1. Express in exponential form:

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


`282.3^@ = 4.93` radians


`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:


`-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`


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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