Partial Fractions Expansion of Cotangent/Proof 3

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Theorem

Let $x \in \R \setminus \Z$, that is such that $x$ is a real number that is not an integer.

Then:

$\ds \pi \cot \pi x = \dfrac 1 x + 2 x \sum_{n \mathop = 1}^\infty \frac 1 {x^2 - n^2}$


Proof

From Euler's Reflection Formula:

$\forall x \notin \Z: \map \Gamma x \map \Gamma {1 - x} = \dfrac \pi {\map \sin {\pi x} }$

Taking the logarithm of both sides:

\(\ds \map \ln {\map {\Gamma} x } + \map \ln {\map {\Gamma} {1 - x} }\) \(=\) \(\ds \map \ln {\pi } - \map \ln {\map \sin {\pi x} }\) Sum of Logarithms/Natural Logarithm and Difference of Logarithms


Taking the derivative of both sides:

\(\ds \frac {\map {\Gamma'} x} {\map \Gamma x} - \frac {\map {\Gamma'} {1 - x} } {\map \Gamma {1 - x} }\) \(=\) \(\ds -\frac 1 {\map \sin {\pi x} } \map \cos {\pi x} \pi\) Derivative of Composite Function, Derivative of Natural Logarithm Function and Derivative of Sine Function
\(\ds \) \(=\) \(\ds -\pi \map \cot {\pi x}\) Definition of cotangent

We now have:

\(\ds \pi \map \cot {\pi x}\) \(=\) \(\ds \frac {\map {\Gamma'} {1 - x} } {\map \Gamma {1 - x} } - \frac {\map {\Gamma'} x} {\map \Gamma x}\) multiplying both sides by $-1$
\(\ds \) \(=\) \(\ds \paren {-\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {n - x } } } - \paren {-\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {x + n - 1} } }\) Reciprocal times Derivative of Gamma Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \paren {\frac 1 {x + n - 1} - \frac 1 {n - x } }\) Linear Combination of Convergent Series
\(\ds \) \(=\) \(\ds \frac 1 x + \sum_{n \mathop = 1}^\infty \paren {\frac 1 {x + n } - \frac 1 {n - x } }\) reindexing the sum
\(\ds \) \(=\) \(\ds \frac 1 x + \sum_{n \mathop = 1}^\infty \paren {\frac 1 {x + n } + \frac 1 {x - n } }\)
\(\ds \) \(=\) \(\ds \dfrac 1 x + 2 x \sum_{n \mathop = 1}^\infty \frac 1 {x^2 - n^2}\) Difference of Two Squares

$\blacksquare$


Sources

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