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; and
θ is in radians.)
Express `5(cos 135^@ +j\ sin\ 135^@)` in exponential form.
Express `-1 + 5j` in exponential form.
SUMMARY: Forms of a complex number
These expressions have the same value. They are just different ways of expressing the same complex number.
|Rectangular form||Polar form||Exponential form|
|x + yj||`r(cos\ θ + j\ sin\ θ)` `= r\ "cis"\ θ` `= r∠ θ`||rejθ|
1. For Polar Form, θ can be in degrees OR radians.
2. For Exponential Form, θ MUST be in radians.
1. Express in exponential form:
`4.50(cos\ 282.3^@+ j\ sin\ 282.3^@)`
2. Express in exponential form: `-1 - 5j`
3. Express in polar and rectangular forms: `2.50e^(3.84j)`