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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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 18E
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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.
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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