Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha
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Theorem
For $0 < \theta < 2 \pi$:
- $\ds \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \sin {\alpha u} \dfrac {\sin \paren {m + \frac 1 2} u \rd u} {\sin \frac 1 2 u}$
Proof
We have:
\(\ds \sum_{n \mathop = -m}^m e^{i \paren {n + \alpha} \theta}\) | \(=\) | \(\ds \sum_{n \mathop = -m}^m e^{i n \theta} e^{i \alpha \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{i \alpha \theta} e^{-i m \theta} \sum_{n \mathop = 0}^{2 m} e^{i n \theta}\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds e^{i \alpha \theta} e^{-i m \theta} \paren {\dfrac {e^{i \paren {2 m + 1} \theta} - 1} {e^{i \theta} - 1} }\) | Sum of Geometric Sequence | ||||||||||
\(\ds \) | \(=\) | \(\ds e^{i \alpha \theta} e^{-i m \theta} \paren {\dfrac {e^{i \paren {2 m + 1} \theta / 2} \paren {e^{i \paren {2 m + 1} \theta / 2} - e^{-i \paren {2 m + 1} \theta / 2} } } {e^{i \theta / 2} \paren {e^{i \theta / 2} - e^{i \theta / 2} } } }\) | extracting factors | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{i \alpha \theta} \paren {\dfrac {e^{i \paren {2 m + 1} \theta / 2} - e^{-i \paren {2 m + 1} \theta / 2} } {e^{i \theta / 2} - e^{i \theta / 2} } }\) | Exponential of Sum and some algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{i \alpha \theta} \frac {\map \sin {\paren {2 m + 2} \theta / 2} } {\map \sin {\theta / 2} }\) | Euler's Sine Identity | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = -m}^m \paren {\cos \paren {n + \alpha} \theta + i \sin \paren {n + \alpha} \theta}\) | \(=\) | \(\ds \paren {\map \cos {\alpha \theta} + i \map \sin {\alpha \theta} } \frac {\map \sin {\paren {m + \frac 1 2} \theta } } {\map \sin {\theta / 2} }\) | Euler's Formula and simplifying | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = -m}^m \sin \paren {n + \alpha} \theta\) | \(=\) | \(\ds \map \sin {\alpha \theta} \frac {\map \sin {\paren {m + \frac 1 2} \theta} } {\map \sin {\theta / 2} }\) | equating imaginary parts |
Note that the right hand side at $(1)$ is not defined when $e^{i u} = 1$.
This happens when $u = 2 k \pi$ for $k \in \Z$.
For the given range of $0 < \theta < 2 \pi$ it is therefore seen that $(1)$ does indeed hold.
Then:
\(\ds \int_0^\theta \sin \paren {\alpha + n} u \rd u\) | \(=\) | \(\ds \intlimits {\dfrac {-\cos \paren {n + \alpha} u} {n + \alpha} } {u \mathop = 0} {u \mathop = \theta}\) | Primitive of $\sin a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {-\cos \paren {n + \alpha} \theta} {n + \alpha} } - \paren {\dfrac {-\cos \paren {n + \alpha} 0} {n + \alpha} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha}\) | Cosine of Zero is One | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha}\) | \(=\) | \(\ds \sum_{n \mathop = -m}^m \int_0^\theta \sin \paren {\alpha + n} u \rd u\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\theta \sum_{n \mathop = -m}^m \sin \paren {\alpha + n} u \rd u\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\theta \map \sin {\alpha u} \frac {\map \sin {\paren {m + \frac 1 2} u} } {\map \sin {u / 2} } \rd u\) | from $(2)$, changing the variable name |
Hence the result.
$\blacksquare$
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Exercises on Chapter $\text {II}$: $2$.