Arbitrary Power of Complex Number

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Theorem

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

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

Then:

\(\ds z^n\) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j a^{n - j} b^j}\)
\(\ds \) \(=\) \(\ds \paren {a^n - \dbinom n 2 a^{n - 2} b^2 + \dbinom n 4 a^{n - 4} b^4 - \cdots}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \paren {\dbinom n 1 a^{n - 1} b - \dbinom n 3 a^{n - 3} b^3 + \cdots}\)


Proof

Lemma

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:

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

$\Box$


The proof proceeds by induction.

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:

$\ds z^n = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j a^{n - j} b^j}$


$\map P 1$ is the case:

\(\ds z^1\) \(=\) \(\ds \paren {a + i b}^1\)
\(\ds \) \(=\) \(\ds \dbinom 1 0 a^1 b^0 + i \dbinom 1 1 a^0 b^1\)
\(\ds \) \(=\) \(\ds \dbinom 1 0 a^{1 - 0} b^0 + i \dbinom 1 1 a^{1 - 1} b^1\)
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le 1 \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom 1 j a^{1 - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom 1 j a^{1 - j} b^j}\)

Thus $\map P 1$ is seen to hold.


Basis for the Induction

$\map P 2$ is the case:

\(\ds z^2\) \(=\) \(\ds \paren {a + i b}^2\)
\(\ds \) \(=\) \(\ds a^2 - b^2 + i \paren {2 a b}\) Square of Complex Number
\(\ds \) \(=\) \(\ds \dbinom 2 0 a^{2 - 0} b^0 + \dbinom 2 2 a^{2 - 2} b^2 + i \dbinom 1 1 a^{2 - 1} b^1\)
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le 2 \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom 2 j a^{2 - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le 2 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom 2 j a^{2 - j} b^j}\)


Thus $\map P 2$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is the induction hypothesis:

$\ds z^k = \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j} b^j}$


from which it is to be shown that:

$\ds z^{k + 1} = \paren {\sum_{\substack {0 \mathop \le j \mathop \le k + 1 \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le k + 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j}$


Induction Step

This is the induction step:


From Lemma:

\(\ds z^{k + 1}\) \(=\) \(\ds u_{k + 1} + i v_{k + 1}\)
\(\ds \) \(=\) \(\ds \paren {a u_k - b v_k} + i \paren {a v_k + b u_k}\)

where $z^k = u_k + i v_k$.


Taking the real part:

\(\ds u_{k + 1}\) \(=\) \(\ds a \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j} b^j} - b \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j} b^j}\) by Induction Hypothesis
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j + 1} b^j} - \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j} b^{j + 1} }\)
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j + 1} b^j} + \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {\paren {j - 1} / 2} + 1} \dbinom k j a^{k - j} b^{j + 1} }\)
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j + 1} b^j} + \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j + 1} / 2} \dbinom k j a^{k - j} b^{j + 1} }\) simplification
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j + 1} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k + 1 \\ \text {$j$ even} } } \paren {-1}^{\paren {\paren {j + 1} - 1} / 2} \dbinom k {j - 1} a^{k - \paren {j - 1} } b^j}\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k + 1 - j} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k + 1 \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k {j - 1} a^{k + 1 - j} b^j}\) simplifying
\(\ds \) \(=\) \(\ds a^{k + 1} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k + 1 - j} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k {j - 1} a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ is even} } b^{k + 1}\) where $\sqbrk \cdots$ is Iverson's convention
\(\ds \) \(=\) \(\ds a^{k + 1} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \paren {\dbinom k j + \dbinom k {j - 1} } a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ is even} } b^{k + 1}\) General Distributivity Theorem
\(\ds \) \(=\) \(\ds a^{k + 1} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ is even} } b^{k + 1}\) Pascal's Rule
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le {k + 1} \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j}\) as $\dbinom {k + 1} 0 = \dbinom {k + 1} {k + 1} = 1$


Taking the imaginary part:

\(\ds v_{k + 1}\) \(=\) \(\ds a \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j} b^j} + b \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j} b^j}\) by Induction Hypothesis
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j + 1} b^j} + \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom k j a^{k - j} b^{j + 1} }\)
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k - j + 1} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k + 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k {j - 1} a^{k - j - 1} b^j}\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k + 1 - j} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k + 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k {j - 1} a^{k + 1 - j} b^j}\) simplifying
\(\ds \) \(=\) \(\ds \sqbrk {\text {$0$ is odd} } a^{k + 1} \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k j a^{k + 1 - j} b^j} + \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom k {j - 1} a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ odd} } b^{k + 1}\) where $\sqbrk \cdots$ is Iverson's convention
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \paren {\dbinom k j + \dbinom k {j - 1} } a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ odd} } b^{k + 1}\) General Distributivity Theorem, and of course $\sqbrk {\text {$0$ is odd} } = 0$
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {1 \mathop \le j \mathop \le k \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ odd} } b^{k + 1}\) Pascal's Rule
\(\ds \) \(=\) \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le k + 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j} + \sqbrk {\text {$k + 1$ odd} } b^{k + 1}\) $0$ vacuously


Thus we have shown:

$\ds z^{k + 1} = \paren {\sum_{\substack {0 \mathop \le j \mathop \le k + 1 \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le k + 1 \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom {k + 1} j a^{k + 1 - j} b^j}$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \Z_{\ge 0}: \ds z^n = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j a^{n - j} b^j}$

$\blacksquare$


Sources

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