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Review of complex numbers 4-R₁e -R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. C8=2mi/8 2122=R₁ R₂+8) The complex conjugate of z = Rei=a+bi is Caz-Ro40+2x/8) z = Re=a-bi, which is the reflection of z across the real axis. Note that |z2=z-7= Re Re = R²e = R² ⇒ z=√√zz=√√a2+ b² = R. Explore the connections between potential theory and complex analysis. 1. Green's Functions in Complex Domains: ⚫a. Define the Green's function for a domain DCC with a pole at CED. ⚫ b. Construct the Green's function for the unit disk D and verify its properties. 2. Poisson Kernel and Harmonic Functions: ⚫a, Derive the Poisson kernel for the upper half-plane H = {C | 3() > 0}. ⚫b. Use the Poisson kernel to solve the Dirichlet problem for H with prescribed boundary values on the real axis. 3. Harmonic Measure and Conformal Mappings: ⚫a. Define the harmonic measure w(3, E, D) for a subset ECOD and a point z € D. ⚫b. Show how conformal mappings preserve harmonic measures and use this property to compute harmonic measures in transformed domains. 4. Capacity and Extremal Problems: ⚫a. Define the conformal capacity of a compact set KCC. ⚫ b. Solve an extremal problem to find the conformal capacity of a circle of radius centered at the origin. 5. Potential Theory and Analytic Functions: ⚫a. Explain the relationship between subharmonic functions and analytic functions. ⚫b. Prove that the logarithm of the modulus of a non-constant analytic function is a subharmonic function.

Review of complex numbers 4-R₁e -R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. C8=2mi/8 2122=R₁ R₂+8) The complex conjugate of z = Rei=a+bi is Caz-Ro40+2x/8) z = Re=a-bi, which is the reflection of z across the real axis. Note that |z2=z-7= Re Re = R²e = R² ⇒ z=√√zz=√√a2+ b² = R. Explore the connections between potential theory and complex analysis. 1. Green's Functions in Complex Domains: ⚫a. Define the Green's function for a domain DCC with a pole at CED. ⚫ b. Construct the Green's function for the unit disk D and verify its properties. 2. Poisson Kernel and Harmonic Functions: ⚫a, Derive the Poisson kernel for the upper half-plane H = {C | 3() > 0}. ⚫b. Use the Poisson kernel to solve the Dirichlet problem for H with prescribed boundary values on the real axis. 3. Harmonic Measure and Conformal Mappings: ⚫a. Define the harmonic measure w(3, E, D) for a subset ECOD and a point z € D. ⚫b. Show how conformal mappings preserve harmonic measures and use this property to compute harmonic measures in transformed domains. 4. Capacity and Extremal Problems: ⚫a. Define the conformal capacity of a compact set KCC. ⚫ b. Solve an extremal problem to find the conformal capacity of a circle of radius centered at the origin. 5. Potential Theory and Analytic Functions: ⚫a. Explain the relationship between subharmonic functions and analytic functions. ⚫b. Prove that the logarithm of the modulus of a non-constant analytic function is a subharmonic function.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 1ECP: Plot z=34i in the complex plane and find its absolute value.
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Review of complex numbers 4-R₁e -R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. C8=2mi/8 2122=R₁ R₂+8) The complex conjugate of z = Rei=a+bi is Caz-Ro40+2x/8) z = Re=a-bi, which is the reflection of z across the real axis. Note that |z2=z-7= Re Re = R²e = R² ⇒ z=√√zz=√√a2+ b² = R. Explore the connections between potential theory and complex analysis. 1. Green's Functions in Complex Domains: ⚫a. Define the Green's function for a domain DCC with a pole at CED. ⚫ b. Construct the Green's function for the unit disk D and verify its properties. 2. Poisson Kernel and Harmonic Functions: ⚫a, Derive the Poisson kernel for the upper half-plane H = {C | 3() > 0}. ⚫b. Use the Poisson kernel to solve the Dirichlet problem for H with prescribed boundary values on the real axis. 3. Harmonic Measure and Conformal Mappings: ⚫a. Define the harmonic measure w(3, E, D) for a subset ECOD and a point z € D. ⚫b. Show how conformal mappings preserve harmonic measures and use this property to compute harmonic measures in transformed domains. 4. Capacity and Extremal Problems: ⚫a. Define the conformal capacity of a compact set KCC. ⚫ b. Solve an extremal problem to find the conformal capacity of a circle of radius centered at the origin. 5. Potential Theory and Analytic Functions: ⚫a. Explain the relationship between subharmonic functions and analytic functions. ⚫b. Prove that the logarithm of the modulus of a non-constant analytic function is a subharmonic function.
Transcribed Image Text:Review of complex numbers 4-R₁e -R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. C8=2mi/8 2122=R₁ R₂+8) The complex conjugate of z = Rei=a+bi is Caz-Ro40+2x/8) z = Re=a-bi, which is the reflection of z across the real axis. Note that |z2=z-7= Re Re = R²e = R² ⇒ z=√√zz=√√a2+ b² = R. Explore the connections between potential theory and complex analysis. 1. Green's Functions in Complex Domains: ⚫a. Define the Green's function for a domain DCC with a pole at CED. ⚫ b. Construct the Green's function for the unit disk D and verify its properties. 2. Poisson Kernel and Harmonic Functions: ⚫a, Derive the Poisson kernel for the upper half-plane H = {C | 3() > 0}. ⚫b. Use the Poisson kernel to solve the Dirichlet problem for H with prescribed boundary values on the real axis. 3. Harmonic Measure and Conformal Mappings: ⚫a. Define the harmonic measure w(3, E, D) for a subset ECOD and a point z € D. ⚫b. Show how conformal mappings preserve harmonic measures and use this property to compute harmonic measures in transformed domains. 4. Capacity and Extremal Problems: ⚫a. Define the conformal capacity of a compact set KCC. ⚫ b. Solve an extremal problem to find the conformal capacity of a circle of radius centered at the origin. 5. Potential Theory and Analytic Functions: ⚫a. Explain the relationship between subharmonic functions and analytic functions. ⚫b. Prove that the logarithm of the modulus of a non-constant analytic function is a subharmonic function.
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