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A singular point of an
algebraic curve is a point where the curve has "nasty" behavior such as a
cusp or a point of self-intersection (when the underlying field
K is taken as the
reals). More formally, a point (
a, b) on a curve
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is singular if the
x and
y partial derivatives of
f are both zero at the point (
a, b). (If the field
K is not the
reals or
complex numbers, then the
partial derivative is computed formally using the usual rules of
calculus.)
Consider the following two examples. For the curve
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(1)
the
cusp at (0, 0) is a singular point. For the curve
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(2)
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is a nonsingular point and this curve is nonsingular.
For a
second-order ordinary differential equation, consider
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(3)
If
P(x) and
Q(x) remain
finite at
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, then
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is called an
ordinary point . If either
P(x) or
Q(x) diverges as
http://mathworld.wolfram.com/s1img1155.gif , then
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is called a singular point. Singular points are further classified as follows:
1. If either
P(x) or
Q(x) diverges as
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but
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and
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remain
finite as
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, then
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is called a
regular singular point (or
nonessential singularity).
2. If
P(x) diverges more quickly than
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, so
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approaches
infinity as
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, or
Q(x) diverges more quickly than
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so that
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goes to
infinity as
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, then
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is called an
irregular singularity (or
essential singularity).
Singular points are sometimes known as
singularities, and vice versa.
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