Power Series of Sine of Odd Theta
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Theorem
Let $r \in \R$ such that $\size r < 1$.
Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
\(\ds \sum_{k \mathop \ge 0} \map \sin {2 k + 1} \theta r^k\) | \(=\) | \(\ds \sin \theta + r \sin 3 \theta + r^2 \sin 5 \theta + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}\) |
Proof
From Euler's Formula:
- $\map \exp {i \theta} = \cos \theta + i \sin \theta$
Hence:
\(\ds \sum_{k \mathop = 0}^\infty \map \sin {2 k + 1} \theta r^k\) | \(=\) | \(\ds \map \Im {\sum_{k \mathop = 0}^\infty r^k \map \exp {\paren {2 k + 1} i \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\map \exp {i \theta} \sum_{k \mathop = 0}^\infty \paren {r \map \exp {2 i \theta} }^k}\) | as $\map \Im {\exp \paren {i \times 0 \times x} } = \map \Im 1 = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\map \exp {i \theta} } {1 - r \map \exp {2 i \theta} } }\) | Sum of Infinite Geometric Sequence: valid because $\size r < 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\map \exp {i \theta} \paren {1 - r \map \exp {-2 i \theta} } } {\paren {1 - r \map \exp {2 i \theta} } \paren {1 - r \map \exp {-2 i \theta} } } }\) | multiplying by 1 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\map \exp {i \theta} \paren {1 - r \map \exp {-2 i \theta} } } {1 - r \paren {\map \exp {-2 i \theta} + \map \exp {2 i \theta} } + r^2} }\) | expanding | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\map \exp {i \theta} \paren {1 - r \map \exp {-2 i \theta} } } {1 - 2 r \cos 2 \theta + r^2} }\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\paren {\cos \theta + i \sin \theta} \paren {1 - r \paren {\cos 2 \theta - i \sin 2 \theta} } } {1 - 2 r \cos 2 \theta + r^2} }\) | Euler's Formula, Corollary to Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\cos \theta + i \sin \theta - r \cos \theta \cos 2 \theta + i r \cos \theta \sin 2 \theta - i r \sin \theta \cos 2 \theta - r \sin \theta \sin 2 \theta} {1 - 2 r \cos 2 \theta + r^2} }\) | expanding | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin \theta + r \cos \theta \sin 2 \theta - r \sin \theta \cos 2 \theta} {1 - 2 r \cos 2 \theta + r^2}\) | taking imaginary part | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin \theta + r \paren {\sin 2 \theta \cos \theta - \cos 2 \theta \sin \theta} } {1 - 2 r \cos 2 \theta + r^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin \theta + r \map \sin {2 \theta - \theta} } {1 - 2 r \cos 2 \theta + r^2}\) | Sine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {1 + r } \sin \theta} {1 - 2 r \cos 2 \theta + r^2}\) | after simplification |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $10$