Evaluating a Fractional Derivative of $cot(x)$

Question

First, we should notice that $$\sum_{m=1}^{\infty}\frac{1}{x-\pi n} +\sum_{n=0}^{\infty}\frac{1}{x+\pi n}=\cot (x)$$ (I can no longer find the proof of this, sorry) and that if we take the $a$'th derivative of both sides we get $$\Gamma(a+1)(-1)^a[\sum_{m=1}^{\infty}\frac{1}{(x-\pi n)^{a+1}}+\sum_{n=0}^{\infty}\frac{1}{(x+\pi n)^{a+1}}]=\frac{d^a}{dx^a}(\cot x)$$ This is a definition of a fractional derivative of $cot(x)$, but the question I have is does this work for all real numbers? I know it works for rational numbers that have an odd denominator. Thank you.