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Powers of Complex Numbers
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Using De Moivre's Theorem to find powers of complex numbers in polar form, z=reiθ, so that zn=rneinθ=rncis(nθ),
Powers of complex numbers
AHL AA 1.14
Now that we know how to represent complex numbers in the form reiθ, we can understand De Moivre's Theorem, which gives us a shortcut to finding powers of complex numbers:
z=reiθ⇒zn=(reiθ)n=rneinθ
And since reiθ=rcisθ:
[r(cosθ+isinθ)]n=(rcisθ)n=rneinθ=rncis(nθ)📖
Roots of complex numbers
AHL AA 1.14
De Moivre's Theorem can also be used to find the nth roots of complex numbers:
n√reiθ=(reiθ)1/n=n√r⋅eiθ/n🚫
or equivalently
n√rcisθ=n√rcis(nθ)🚫
However, since cisθ=cis(θ+2kπ) for any k∈Z, then we actually have
n√rcisθ=n√rcis(nθ+2kπ),k=0,1…n−1🚫
Note that k stops at n−1 since when k=n we have
cis(nθ+2nπ)=cis(θ+2π)=cisθ
Nice work completing Powers of Complex Numbers, here's a quick recap of what we covered:
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