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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 4th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded an Idea-paragraph to _[[non-perturbative effect]]_, also to _[perturbation theory - divergence/convergence](http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence)_.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2018
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   Author: Urs
   Format: MarkdownItexstarted a note on worldsheet and brane instantons ([here](https://ncatlab.org/nlab/show/non-perturbative+effect#WorldsheetAndBraneInstantons)) but not optimized for public consumption yet <a href="https://ncatlab.org/nlab/revision/diff/non-perturbative+effect/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/non-perturbative+effect/21">v21</a>, <a href="https://ncatlab.org/nlab/show/non-perturbative+effect">current</a>

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

   diff, v21, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 19th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](https://arxiv.org/abs/1102.2380), [ISBN 978-86-82441-30-4](http://www.proceedings.com/22646.html), [book webpage](http://www.mphys6.ipb.ac.rs/proceedings6.htm), [book TOC pdf](http://toc.proceedings.com/22646webtoc.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/non-perturbative+effect/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/non-perturbative+effect/37">v37</a>, <a href="https://ncatlab.org/nlab/show/non-perturbative+effect">current</a>

   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)

   diff, v37, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2021
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   Author: Urs
   Format: MarkdownItexadded pointer to: * Mario Flory, [[Robert C. Helling]], Constantin Sluka, Section 2 of: *How I Learned to Stop Worrying and Love QFT* ([arXiv:1201.2714](https://arxiv.org/abs/1201.2714)) <a href="https://ncatlab.org/nlab/revision/diff/non-perturbative+effect/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/non-perturbative+effect/40">v40</a>, <a href="https://ncatlab.org/nlab/show/non-perturbative+effect">current</a>

   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)

   diff, v40, current

5. • CommentRowNumber5.
   • CommentAuthorGuest
   • CommentTimeApr 6th 2023
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   Author: Guest
   Format: MarkdownItexStrictly 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.

   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.