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modif singularities

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Pierre-Francois Loos 2020-11-29 20:53:43 +01:00
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@ -196,7 +196,8 @@ We refer the interested reader to the excellent book of Moiseyev for a general o
At a singular point, a function and/or its derivatives becomes infinite or undefined (hence non analytic). 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. 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 which are lines or curves 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. Critical points are singularities which lie on the real axis and where the nature of the function undergoes a sudden transition.
However, these do not clearly belong to a given class of singularities and they cannot be rigorously classified as they have more complicated functional forms.
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