Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 1=R₁e z=Re The complex conjugate of z = Rei=a+bi is Caz-Re(+2x/8) z= Re=a-bi, which is the reflection of z across the real axis. Note that 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. Ca=2mi/8 Iz zz Re Re-i = R2e0 = R2 |z|=√√zz=√√√² + b²=R. Let f (z) be a function holomorphic in an annulus A = {e Cr<< R}, where 0<
Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 1=R₁e z=Re The complex conjugate of z = Rei=a+bi is Caz-Re(+2x/8) z= Re=a-bi, which is the reflection of z across the real axis. Note that 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. Ca=2mi/8 Iz zz Re Re-i = R2e0 = R2 |z|=√√zz=√√√² + b²=R. Let f (z) be a function holomorphic in an annulus A = {e Cr<< R}, where 0<
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 2ECP
Question

Transcribed Image Text:Review of complex numbers
22-R₂e
2122 R₁ R₂+82)
01+02
1=R₁e
z=Re
The complex conjugate of z = Rei=a+bi is
Caz-Re(+2x/8)
z= Re=a-bi,
which is the reflection of z across the real axis.
Note that
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.
Ca=2mi/8
Iz zz Re Re-i = R2e0 = R2
|z|=√√zz=√√√² + b²=R.
Let f (z) be a function holomorphic in an annulus A = {e Cr<< R}, where 0<
<R≤0.
1. Laurent Series Expansion:
a. Prove that f(z) can be represented as a Laurent series around zo within the annulus A,
that is,
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 (2) in the annulus
A.
2. Classification of Singularities:
a. Using the Laurent series, classify the singularity of f(z) 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(2) at an isolated singularity zo and express it in terms of the
Laurent series coefficients.
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