First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse phenomena together under a single governing principle.
Now that we have developed some machinery for power series, we are ready to connect them to the earlier chapters. Our first big goal in this chapter is to prove that every power series represents a holomorphic function in its disk of convergence and, vice versa, that every holomorphic function can be locally represented by a power series.
Looking back at Figure 7.0.1 suggests that we expand the function \(\frac{ \exp(z) }{ \sin(z) }\) locally into something like power series centered at 0 and \(\pi\text{,}\) and with any luck we can then use Proposition 7.3.6 to integrate. Of course, \(\frac{ \exp(z) }{ \sin(z) }\) has singularities at 0 and \(\pi\text{,}\) so there is no hope of computing power series at these points. We will develop an analogue of a power series centered at a singularity.
