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), and applying the argument rule arg(zw)=arg(z)+arg(w).
Powers of complex numbers
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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)📖
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