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In mathematics, a singularity is in general a point at which
a given mathematical object is not defined or lacks some "nice" property, such as differentiability. See singularity theory for
general discussion.
For example, the function
- f(x) = 1/x
has a singularity at x = 0, where it explodes to ±∞ and isn't defined. The function g(x) =
|x| (see absolute value) also has a singularity at x
= 0, since it isn't differentiable there. The algebraic set defined by y2 = x2 in the
(x,y) coordinate system has a singularity at (0,0) because it doesn't admit a tangent there. The algebraic set defined by y2 = x also has a singularity at (0,0),
this time because it has a "corner" at that point.
In complex analysis, we distinguish between four kinds of
singularity. Suppose U is an open subset of C, a is an element of U and f is a holomorphic function defined on U-{a}.
- the point a is a removable singularity of
f if there exists a holomorphic function g defined on all of U such that
f(z)=g(z) for all z in U-{a}.
- the point a is a pole of f
if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z -
a)n for all z in U-{a}.
- A branch point of f is one requiring a more verbose
definition; see the article of that title.
See also
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