Review of complex numbers 22-R₂e 81+02 2122=R₁₂(+2) 4-R₁e z=Re C82=Ro(+218) Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB=2mi/8 Let f (2) be a function holomorphic in an annulus A = {ze Cr<|z0|< R}, where 0<
Review of complex numbers 22-R₂e 81+02 2122=R₁₂(+2) 4-R₁e z=Re C82=Ro(+218) Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB=2mi/8 Let f (2) be a function holomorphic in an annulus A = {ze Cr<|z0|< R}, where 0<
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 18E
Question

Transcribed Image Text:Review of complex numbers
22-R₂e
81+02
2122=R₁₂(+2)
4-R₁e
z=Re
C82=Ro(+218)
Do not use AI, I need real solution, attach required graph and code wherever needed.
For reference I have attached the image, but if you need any reference then check out the book by
Churchill only.
CB=2mi/8
Let f (2) be a function holomorphic in an annulus A = {ze Cr<|z0|< R}, where 0<
<R<∞.
1. Laurent Series Expansion:
⚫a. Prove that f(z) can be represented as a Laurent series around ze within the annulus A,
that is,
The complex conjugate of z = Rei=a+bi is
z= Re = a - bi,
which is the reflection of z across the real axis.
Note that
z=zz Re Re-i =R2e0 = R2 =
z=√√zz √√²+b² = R.
=
f(=) = Σan (= -20)".
where the coefficients an are given by
and C is a positively oriented, simple closed contour within A.
⚫b. Demonstrate the uniqueness of the Laurent series representation for f(2) in the annulus
A.
2. Classification of Singularities:
⚫ a. Using the Laurent series, classify the singularity of f(x) at zo as removable, a pole, or an
essential singularity based on the behavior of the coefficients a
⚫b. Provide examples of functions exhibiting each type of singularity and illustrate their
Laurent series expansions.
3. Residue Calculation:
⚫a. Define the residue of f() at an isolated singularity zo and express it in terms of the
Laurent series coefficients.
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