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;
θ is in radians; and
`j=sqrt(-1).`
Example 1
Express `5(cos 135^@ +j\ sin\ 135^@)` in exponential form.
Answer
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.
Answer
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.
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
rejθ
θ MUST be in radians for Exponential form.
Exercises
1. Express in exponential form:
`4.50(cos\ 282.3^@+ j\ sin\ 282.3^@)`
Answer
`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`
Answer
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)`
Answer
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:
