Some power series

May 20, 2020 — ~redtrumpet

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