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Some power series

Today I studied genera, which provide a way to produce toopological invariants of manifolds. Genera are intimately related to invertible power series which start with $1 + \dotsb$. For example the common genus $L$-genus belongs to the power series of $\frac{x}{\tanh(x)}.$ The so called Â-genus looks even more intimidating, it belongs to the power series of $\frac{\sqrt{x}/2}{\tanh(\sqrt{x}/2)}.$ So this leaves the question: Why are those functions above analytic? Formally, they are not even defined at $x = 0$, because you end up with $\frac{0}{0}$. Nevertheless both functions are indeed defined at $x = 0$, and both are analytic, meaning that you can write them as a power series $\sum_{k=0}^\infty c_k x^k$ for certain coefficients $c_k$. I want to explain how to derive the power series representation, i.e. how to compute the $c_k$ for both series.

Let's start with unrolling the definitions: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)},$ and $\cosh$ and $\sinh$ are defined by $\sinh = \frac{1}{2}(e^x - e^{-x}) \text{ and } \cosh = \frac{1}{2}(e^x + e^{-x}).$ Using the well-known series expansion of $e^x$, we can write $\sinh = \frac{1}{2}\left(\sum_{k=0}^\infty \frac{x^k}{k!} - \sum_{k=0}^\infty (-1)^k \frac{x^k}{k!}\right) = \frac{1}{2} \left(\sum_k \frac{x^{2k+1}}{(2k+1)!} \right)$ and $\cosh = \frac{1}{2}\left(\sum_{k=0}^\infty \frac{x^k}{k!} + \sum_{k=0}^\infty (-1)^k \frac{x^k}{k!}\right) = \frac{1}{2} \left(\sum_k \frac{x^{2k}}{(2k)!} \right).$ The next stop is to compute the

tags: math