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## Discussion Tag Cloud

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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 4th 2013
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 28th 2018

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 19th 2020

added more publication data and links to this item:

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021

• CommentRowNumber5.
• CommentAuthorGuest
• CommentTimeApr 6th 2023

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 12th 2023

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)