Download 1 Exponential Form 2 Quiz 26A

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MA 1165 - Lecture 26
4/22/09
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Exponential Form
As we have seen, if we have two complex numbers in polar form
z1 = r1 ( cos(θ1 ) + i sin(θ1 ) )
(1)
z2 = r2 ( cos(θ2 ) + i sin(θ2 ) ) ,
(2)
z1 · z2 = r1 · r2 ( cos(θ1 + θ2 ) + i sin(θ1 + θ2 ) ) .
(3)
and
and we multiply them together, we get
One thing you may notice is that a multiplication has become, at least in part, an addition. We’ve seen this
before with the exponential functions. In particular,
ea · eb = ea+b .
(4)
So what should
ea+bi
(5)
equal? Well, it doesn’t have to equal anything in particular, but if we want to extend the exponential
function to the complex numbers, and we want this function to retain its basic properties, then it pretty
much has to be defined so that
ea+bi = ea · ebi = ea (cos(b) + i sin(b)).
(6)
In particular,
eiθ = cos(θ) + i sin(θ).
When we do this, our polar form for complex numbers becomes
z = reiθ .
(7)
(8)
This gives us a more compact polar form, but it also brings the exponential functions and the trig functions
together. Any sort of “unification” is an indication that we’ve done something right. It also makes DeMoivre’s
theorem easy to remember.
Keep in mind that this is a purely mathematical definition.
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Quiz 26A
Write the following complex numbers in the new (exponential) polar form.
1.
i.
2.
−3.
3.
1 + i.
4.
3 − 2i.
Write the following complex form in the regular (rectangular) form.
5.
3eπi .
6.
2eπi/2 .
7.
eπ/3 .
MA 1165 - Lecture 26
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No Homework 26
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