Fourier Series for Logarithm of Sine of x over 0 to Pi/Proof 2

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Theorem

$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$

where $0 < x < \pi$.


Proof

\(\ds \sum_{n \mathop = 1}^\infty \dfrac {\cos 2 n x} n\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\map \exp {2 i n x} + \map \exp {-2 i n x} } {2 n}\) Euler's Cosine Identity: $\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$
\(\ds \) \(=\) \(\ds \frac 1 2 \sum_{n \mathop = 1}^\infty \paren {\frac {\paren {\map \exp {2 i x} }^n} n + \frac {\paren {\map \exp {-2 i x} }^n} n}\) Power of Power
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {1 - \map \exp {2 i x} } + \map \ln {1 - \map \exp {-2 i x} } }\) Power Series Expansion for Logarithm of 1 + x: Corollary: $-\map \ln {1 - x} = \ds \sum_{n \mathop = 1}^\infty \dfrac {x^n} n$
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {1 - \map \exp {-2 i x} - \map \exp {2 i x} + 1} }\) Sum of Logarithms
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {2 - 2 \paren {\frac {\map \exp {-2 i x} + \map \exp {2 i x} } 2} } }\) simplifying and multiplying top and bottom by $2$
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {2 - 2 \map \cos {2 x} } }\) Euler's Cosine Identity: $\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {4 \sin^2 x} }\) Double Angle Formula for Cosine: Corollary $2$: $1 - \cos 2 \theta = 2 \sin^2 \theta$
\(\ds \) \(=\) \(\ds -\paren {\map \ln {4 \sin^2 x }^{\frac 1 2} }\) Logarithm of Power
\(\ds \) \(=\) \(\ds -\paren {\map \ln {2 \sin x} }\)
\(\ds \) \(=\) \(\ds -\paren {\ln 2 + \map \ln {\sin x} }\) Sum of Logarithms
\(\ds \leadsto \ \ \) \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\cos 2 n x} n\) \(=\) \(\ds -\ln 2 - \map \ln {\sin x}\)
\(\ds \leadsto \ \ \) \(\ds \map \ln {\sin x}\) \(=\) \(\ds -\ln 2 - \sum_{n \mathop = 1}^\infty \dfrac {\cos 2 n x} n\) rearranging

$\blacksquare$


Sources

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