Difference of Even Powers of z + a and z - a
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Theorem
Let $m \in \Z$ be an integer such that $m > 1$.
Then for all complex number $z$:
- $\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \ds \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$
Proof
From Factors of Difference of Two Even Powers:
- $x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \ds \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$
Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get:
\(\ds \) | \(\) | \(\ds \paren {z + a}^{2 m} - \paren {z - a}^{2 m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {z + a} - \paren {z - a} } \paren {\paren {z + a} + \paren {z - a} } \prod_{k \mathop = 1}^{m - 1} \paren {\paren {z + a}^2 - 2 \paren {z + a} \paren {z - a} \cos \frac {k \pi} m + \paren {z - a}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 a} \paren {2 z} \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + 2 a z + a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m + \paren {z^2 - 2 a z + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 + 2 a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {1 - \cos \frac {k \pi} m} + 2 a^2 \paren {1 + \cos \frac {k \pi} m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {2 \sin^2 \frac {k \pi} {2 m} } + 2 a^2 \paren {2 \cos^2 \frac {k \pi} {2 m} } }\) | Double Angle Formula for Cosine: Corollary $1$ and Double Angle Formula for Cosine: Corollary $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} 4 \sin^2 \frac {k \pi} {2 m} }\) | Definition of Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \map \sin {\pi - \frac {k \pi} {2 m} } }\) | Sine of Supplementary Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {\paren {2 m - k} \pi} {2 m} } \sin \frac {m \pi} {2 m}\) | Sine of Right Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^m \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = m + 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }\) | Translation of Index Variable of Product | |||||||||||
\(\ds \) | \(=\) | \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\frac {2 m} {2^{2 m - 1} } }\) | Product of Sines of Fractions of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 m a z \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} }\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $146$