This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.D Supplementary Sections
D.1 Reminder from Complex Analysis
The analytic function theory is the most powerful tool in the operator
theory. Here we briefly recall few facts of complex analysis used in
this course. Use any decent textbook on complex variables for a
concise exposition. The only difference with our version that we
consider function f(z) of a complex variable z taking value in
an arbitrary normed space V over the field
ℂ. By the direct
inspection we could check that all standard proofs of the listed
results work as well in this more general case.
Definition 1
A function f(
z)
of a complex variable z taking value in a
normed vector space V is called differentiable
at a point
z0 if the following limit (called derivative
of
f(
z)
at z0) exists:
Definition 2
A function f(
z)
is called holomorphic
(or
analytic
) in an
open set Ω⊂ℂ
it is differentiable at any
point of Ω
.
Theorem 3 (Laurent Series)
Let a function f(
z)
be analytical in the annulus r<
z<
R
for some real r<
R, then it could be uniquely represented by the
Laurent series:
|
f(z)= | | ck zk, for some
ck∈ V.
(109) |
Theorem 4 (Cauchy–Hadamard)
The radii r′
and R′
, (r′<
R′
) of convergence of the Laurent
series (109) are given by
|
r′= | | ⎪⎪
⎪⎪ | cn | ⎪⎪
⎪⎪ | 1/n
and
| | = | | ⎪⎪
⎪⎪ | cn | ⎪⎪
⎪⎪ | 1/n.
(110) |
Last modified: February 16, 2025.