added a `tikz`

-graphics (here) meant to illustrate that the non-perturbative region of parameter space is large (really the complement of an infinitesimal subspace)

I’ll probably fine-tune that graphics further tomorrow, and then maybe add it to other related entries (such as to *perturbative QFT*)

Strictly speaking, the $n$-th derivatives for the function $g \mapsto e^{-1/g^2}$ don’t exist at $g = 0$, because $g \mapsto -1/g^2$ and thus $g \mapsto e^{-1/g^2}$ is only defined for $g \gt 0$ or $g \lt 0$. One could analytically continue $g \mapsto e^{-1/g^2}$ and its $n$-th derivatives to $g = 0$ via a piecewise function

$g \mapsto \begin{cases} e^{-1/g^2} & g \gt 0 \vee g \lt 0 \\ 0 & g = 0 \end{cases}$and similarly for its $n$-th derivatives, but in that case, it shouldn’t be surprised that this function’s taylor series expansion at zero is only valid at zero. It’s the same reason why the taylor series expansion for the absolute value function as the identity function on the real numbers is only valid for the positive real numbers, and is inherently due to the piecewise nature of the function.

]]>added pointer to:

- Mario Flory, Robert C. Helling, Constantin Sluka, Section 2 of:
*How I Learned to Stop Worrying and Love QFT*(arXiv:1201.2714)

added more publication data and links to this item:

- Alexander P. Bakulev, Dmitry Shirkov,
*Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory*, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14–23, 2010, Belgrade, Serbia, pp. 27–54 (arXiv:1102.2380, ISBN 978-86-82441-30-4, book webpage, book TOC pdf)

started a note on worldsheet and brane instantons (here) but not optimized for public consumption yet

]]>added an Idea-paragraph to *non-perturbative effect*, also to *perturbation theory - divergence/convergence*.