From b1cd3ef2de3f7c42facbb01b091fe93ce1377c34 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 28 Nov 2020 21:53:41 +0100 Subject: [PATCH 1/2] added small paragraph on singularities --- Manuscript/EPAWTFT.tex | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index b5518aa..8e6b70d 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -191,6 +191,13 @@ More importantly here, although EPs usually lie off the real axis, these singula Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed. We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}} +\titou{Discussion around the different types of singularities in complex analysis. +At a singular point, a function and/or its derivatives becomes infinite or undefined (hence non analytic). +One very common type of singularities (belonging to the family of isolated singularities) are poles where the function behaves $1/(\lambda - \lambda_c)^n$ where $n \in \mathbb{N}^*$ is the order of the pole. +Another class of singularities are branch points resulting from a multi-valued function such as a square root or a logarithm function and usually implying the presence of so-called branch cuts where the function ``jumps'' from one value to another. +Yet another family of singularities are formed by critical points which lie on the real axis, have more complicated functional forms and where the nature of the function undergoes a sudden transition. +} + %%%%%%%%%%%%%%%%%%%%%%% \section{Exceptional Points in Electronic Structure} \label{sec:EPs} From 17ef8dc9c1cf102ab99f86dfbcce27915756df4c Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 28 Nov 2020 22:16:36 +0100 Subject: [PATCH 2/2] added small paragraph on singularities --- Manuscript/EPAWTFT.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 8e6b70d..2783787 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -194,7 +194,7 @@ We refer the interested reader to the excellent book of Moiseyev for a general o \titou{Discussion around the different types of singularities in complex analysis. At a singular point, a function and/or its derivatives becomes infinite or undefined (hence non analytic). One very common type of singularities (belonging to the family of isolated singularities) are poles where the function behaves $1/(\lambda - \lambda_c)^n$ where $n \in \mathbb{N}^*$ is the order of the pole. -Another class of singularities are branch points resulting from a multi-valued function such as a square root or a logarithm function and usually implying the presence of so-called branch cuts where the function ``jumps'' from one value to another. +Another class of singularities are branch points resulting from a multi-valued function such as a square root or a logarithm function and usually implying the presence of so-called branch cuts which are lines or curves where the function ``jumps'' from one value to another. Yet another family of singularities are formed by critical points which lie on the real axis, have more complicated functional forms and where the nature of the function undergoes a sudden transition. }