- Removable singularity
- Essential singularity
- Removable
- Singularity (mathematics)
- Residue theorem
- Residue (complex analysis)
- Classification of discontinuities
- Singularity theory
- Sinc function
- Zeros and poles
removable singularity
Video: removable singularity
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In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
For instance, the (unnormalized) sinc function, as defined by
sinc
(
z
)
=
sin
z
z
{\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}
has a singularity at z = 0. This singularity can be removed by defining
sinc
(
0
)
:=
1
,
{\displaystyle {\text{sinc}}(0):=1,}
which is the limit of sinc as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by sinc being given an indeterminate form. Taking a power series expansion for
sin
(
z
)
z
{\textstyle {\frac {\sin(z)}{z}}}
around the singular point shows that
sinc
(
z
)
=
1
z
(
∑
k
=
0
∞
(
−
1
)
k
z
2
k
+
1
(
2
k
+
1
)
!
)
=
∑
k
=
0
∞
(
−
1
)
k
z
2
k
(
2
k
+
1
)
!
=
1
−
z
2
3
!
+
z
4
5
!
−
z
6
7
!
+
⋯
.
{\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}
Formally, if
U
⊂
C
{\displaystyle U\subset \mathbb {C} }
is an open subset of the complex plane
C
{\displaystyle \mathbb {C} }
,
a
∈
U
{\displaystyle a\in U}
a point of
U
{\displaystyle U}
, and
f
:
U
∖
{
a
}
→
C
{\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} }
is a holomorphic function, then
a
{\displaystyle a}
is called a removable singularity for
f
{\displaystyle f}
if there exists a holomorphic function
g
:
U
→
C
{\displaystyle g:U\rightarrow \mathbb {C} }
which coincides with
f
{\displaystyle f}
on
U
∖
{
a
}
{\displaystyle U\setminus \{a\}}
. We say
f
{\displaystyle f}
is holomorphically extendable over
U
{\displaystyle U}
if such a
g
{\displaystyle g}
exists.
Riemann's theorem
Riemann's theorem on removable singularities is as follows:
The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at
a
{\displaystyle a}
is equivalent to it being analytic at
a
{\displaystyle a}
(proof), i.e. having a power series representation. Define
h
(
z
)
=
{
(
z
−
a
)
2
f
(
z
)
z
≠
a
,
0
z
=
a
.
{\displaystyle h(z)={\begin{cases}(z-a)^{2}f(z)&z\neq a,\\0&z=a.\end{cases}}}
Clearly, h is holomorphic on
D
∖
{
a
}
{\displaystyle D\setminus \{a\}}
, and there exists
h
′
(
a
)
=
lim
z
→
a
(
z
−
a
)
2
f
(
z
)
−
0
z
−
a
=
lim
z
→
a
(
z
−
a
)
f
(
z
)
=
0
{\displaystyle h'(a)=\lim _{z\to a}{\frac {(z-a)^{2}f(z)-0}{z-a}}=\lim _{z\to a}(z-a)f(z)=0}
by 4, hence h is holomorphic on D and has a Taylor series about a:
h
(
z
)
=
c
0
+
c
1
(
z
−
a
)
+
c
2
(
z
−
a
)
2
+
c
3
(
z
−
a
)
3
+
⋯
.
{\displaystyle h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.}
We have c0 = h(a) = 0 and c1 = h'(a) = 0; therefore
h
(
z
)
=
c
2
(
z
−
a
)
2
+
c
3
(
z
−
a
)
3
+
⋯
.
{\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.}
Hence, where
z
≠
a
{\displaystyle z\neq a}
, we have:
f
(
z
)
=
h
(
z
)
(
z
−
a
)
2
=
c
2
+
c
3
(
z
−
a
)
+
⋯
.
{\displaystyle f(z)={\frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+\cdots \,.}
However,
g
(
z
)
=
c
2
+
c
3
(
z
−
a
)
+
⋯
.
{\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots \,.}
is holomorphic on D, thus an extension of
f
{\displaystyle f}
.
Other kinds of singularities
Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types:
In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number
m
{\displaystyle m}
such that
lim
z
→
a
(
z
−
a
)
m
+
1
f
(
z
)
=
0
{\displaystyle \lim _{z\rightarrow a}(z-a)^{m+1}f(z)=0}
. If so,
a
{\displaystyle a}
is called a pole of
f
{\displaystyle f}
and the smallest such
m
{\displaystyle m}
is the order of
a
{\displaystyle a}
. So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles.
If an isolated singularity
a
{\displaystyle a}
of
f
{\displaystyle f}
is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an
f
{\displaystyle f}
maps every punctured open neighborhood
U
∖
{
a
}
{\displaystyle U\setminus \{a\}}
to the entire complex plane, with the possible exception of at most one point.
See also
Analytic capacity
Removable discontinuity
External links
Removable singular point at Encyclopedia of Mathematics