I have been trying to figure out why we simply don't use Taylor Series everywhere to solve integrals. For instance, I have been reading this article Seized opportunities (Notices of the AMS, April 2010), https://www.ams.org/notices/201004/rtx100400476p.pdf . The author says it is difficult to solve some definite integrals, why don't we just use Taylor Series? I don't mean exclusively definite integrals, I may also point out some well-known differential equations, for instance the damped pendulum, which is:
$$\phi^{(2)}(t) + A \phi^{(1)}(t) + B \sin( \phi(t) ) = 0$$
For real $A$ and $B$. Is it because it does not give an analytical answer, in both situations?
EDIT
For instance, let's suppose I have the following complex integral:
$f(z) = \displaystyle\int_{z_0}^z g(s) ds = \displaystyle\int_\Gamma g(s)ds$
Let's also suppose that $g$ is complex analytic and $\Gamma$ is a continuously differentiable path, then could we either:
- Use complex power series to rewrite $g$ and then integrate. (Supposing the sums are interchangeable):
$f(z) = \displaystyle\int_\Gamma \displaystyle\sum_{n=0}^\infty \frac{(s-z_1)^n g^{(n)}(z_1)}{n!} ds$
Or 2) Use the FTC and then rewrite $f$ as a power series:
$f(z) = \displaystyle\sum_{n=0}^\infty \frac{(z-z_1)^n \partial_z^n ( \displaystyle\int_{z_0}^z g(s) ds ) }{n!}$
Are these even possible to do? Additionally, what about the case when it is a improper integral as follows:
$f(z) = \lim_{a \to z_0} \displaystyle\int_a^z g(s) ds$
Could we use analytic power series?