Partial Fractions Expansion of Cotangent/Proof 1

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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

We have that:

$\cot \pi x = \dfrac {\cos \pi x} {\sin \pi x}$

has a denominator which is $0$ at $x = 0, \pm 1, \pm 2, \ldots$.

Hence the limitation on the domain of $x \cot \pi x$ to exclude integer $x$.

Having established that, we should be able to express $\cot \pi x$ in the form:

$\cot \pi x = \dfrac a x + \ds \sum_{n \mathop = 1}^\infty \paren {\frac {b_n} {x - n} + \frac {c_n} {x + n} }$


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using a partial fractions expansion.

By evaluating the coefficients $b_n$ and $c_n$ in the usual manner, they are found to be:


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$\forall n \in \N: b_n = c_n = \dfrac 1 \pi$

The result follows.

$\blacksquare$


Historical Note

The original demonstration of the partial fractions expansion of the cotangent function was presented by Leonhard Paul Euler.

It is non-rigorous, in the sense that it has not at this stage been established that it is in fact possible to perform that expansion of $\cot \pi x$ into such an expansion in the first place.


Sources

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