We talked about cauchy and laurent in the post about Normality of a family (https://kapitals-pi.blogspot.com/2021/02/normality-of-family-repost.html).
Below are there proofs explained.
Given a Laurent series representation of a function
f(z)
in the annulus
rj<|z−a|<rj+1,
where
rj
and
rj+1
are the radii of two consecutive circles in a Laurent arrangement
centered at
a,
the Cauchy formula for Laurent series states:
ck=12Οi∮Cf(z)(z−a)k+1dz
where:
-
ck
is the
k-th
coefficient in the Laurent series expansion of
f(z),
-
C
is a positively oriented contour lying in the annulus
rj<|z−a|<rj+1,
-
k
is any integer. The contour
C
can be any simple closed curve that winds once counterclockwise
around the singularity
a
and lies entirely within the annulus
rj<|z−a|<rj+1. This formula essentially tells us that the coefficient
ck
of
(z−a)k
in the Laurent series expansion of
f(z)
is given by a contour integral around the singularity
a
of
f(z)
divided by
2Οi. This formula is extremely useful in computing coefficients of
Laurent series, especially in cases where finding the coefficients
directly from the series expansion is difficult. Riemann’s theorem typically refers to the Riemann
mapping theorem, which states that any simply connected open subset
of the complex plane that is not the whole plane can be conformally
mapped onto the open unit disk. This theorem is not directly related
to removable singularities. However, if you’re referring to a theorem specifically about
removable singularities, then it’s likely you’re talking about a
result from complex analysis. One of the fundamental theorems
regarding removable singularities is: Theorem (Removable Singularity Theorem): Suppose
f(z)
is holomorphic (analytic) on a punctured neighborhood of
z=a
(i.e., on
0<|z−a|<r)
except possibly at
z=a,
where it has a singularity. If
f(z)
is bounded in a neighborhood of
z=a,
then the singularity at
z=a
is removable. To prove that a function
f(z)
has a removable singularity at
z=0,
one typically demonstrates that the function is holomorphic in some
punctured neighborhood of
z=0
and is bounded in a neighborhood of
z=0.
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