# Taylor Series of Hyperbolic Cotangent Coth(x)

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, it's been two years since I did that and I don't remember it. Is that the only way to find the expansion?

• Well, $\coth$ has a pole in $0$, so there's no Taylor series in $0$. Laurent series is one option, partial fraction decomposition another. What is the goal, what for do you need the expansion? – Daniel Fischer Oct 4 '13 at 18:29
• Langevin Equation. For qualifiers, an old exam question to calculate the polarization of water in an electric field, then get an expression for the permittivity. – walczyk Oct 5 '13 at 2:01

According to http://en.wikipedia.org/wiki/Hyperbolic_function#Taylor_series_expressions, the series form of $\coth x$ is $\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{4^nB_{2n}x^{2n-1}}{(2n)!}$ , where $B_n$ is the $n$th Bernoulli number and only suitable for $0<|x|< \pi$ .
• That is Laurent all right. Since sin n$\pi$ is zero, you would have poles there. – Betty Mock Dec 3 '13 at 4:39