Arbitrary Power of Complex Number/Lemma

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Lemma for Arbitrary Power of Complex Number

Let $z = a + i b$ be a complex number.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $z^n = u_n + i v_n$.

Then $z^{n + 1} = u_{n + 1} + i v_{n + 1}$ where:

\(\ds u_{n + 1}\) \(=\) \(\ds a u_n - b v_n\)
\(\ds v_{n + 1}\) \(=\) \(\ds a v_n + b u_n\)


Proof

\(\ds z^{n + 1}\) \(=\) \(\ds z^n \times z\)
\(\ds \) \(=\) \(\ds \paren {u_n + i v_n} \paren {a + i b}\) Definition of $z$ and $z^n$
\(\ds \) \(=\) \(\ds \paren {a u_n - b v_n} + i \paren {a v_n + b u_n}\) Definition of Complex Multiplication
\(\ds \leadsto \ \ \) \(\ds u_{n + 1}\) \(=\) \(\ds a u_n - b v_n\) Definition of Real Part
\(\, \ds \land \, \) \(\ds v_{n + 1}\) \(=\) \(\ds a v_n + b u_n\) Definition of Imaginary Part

$\blacksquare$


Sources

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