Is every series that matters a Taylor series?

Question

Fourier series are Taylor series in complex z, so I'm wondering if there is any kind of series that represents analytic almost everywhere functions that is foundationally distinct from Taylor series?

Or is every series that matters for analytic functions a Taylor series?

Answer 1

How about the Riemann zeta function? $$ \zeta(s):=\sum_{n\in\mathbb N}n^{-s}\qquad \operatorname{re} s>1 $$

This is manifestly not a Taylor series, yet it is analytic on its domain of definition (and can be continued to all $\mathbb C\setminus\{0\}$).

See also Laurent series, Puiseux series, etc.

Answer 2

Let $f(z) = 1/z$ in $\mathbb C\setminus \{0\}.$ Note $f(z)$ equals the infinite series $f(z) + 0+0+0+\cdots$ everywhere in its domain. But it can't equal a power series in this domain, because such a power series would have to converge in all of $\mathbb C.$ This would imply $f$ has a removable singularity at $0,$ contradiction.

Answer 3

There are also the series $$\sum_{(a,b) \in \mathbb{Z}^2 \setminus\{(0,0)\}} \frac{1}{(a\tau + b)^{2k}},$$ where $k \geq 2$. Though these converge on the open set Im$(\tau) > 0$. They are important in the theories of modular forms and elliptic functions. They satisfy infinitely many symmetries !