Category:Arbitrary Power of Complex Number
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This category contains pages concerning Arbitrary Power of Complex Number:
Let $z = a + i b$ be a complex number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
| \(\ds z^n\) | \(=\) | \(\ds \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j a^{n - j} b^j}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a^n - \dbinom n 2 a^{n - 2} b^2 + \dbinom n 4 a^{n - 4} b^4 - \cdots}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds i \paren {\dbinom n 1 a^{n - 1} b - \dbinom n 3 a^{n - 3} b^3 + \cdots}\) |
Pages in category "Arbitrary Power of Complex Number"
The following 2 pages are in this category, out of 2 total.