Arbitrary Power of Complex Number
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.22$