Review of complex numbers 2122=R₁₂+82) 01+02 1=R₁e 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. Ca-exi/8 R Caz-Re(6+2x/8) Explore the use of Green's functions in solving boundary value problems in complex domains. 1. Green's Function for the Unit Disk: Derive the Green's function G(, C) for the unit disk D = {C||] <1} with a pole at € D. Show that G(:,C) = log || 이는 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 |z2=z.Z= Re Re = R2e0 = R2 => |z|= √√zz=√√a²+b² = R. 2. Green's Function for Multiply Connected Domains: Extend the concept of Green's functions to a doubly connected domain D, such as an annulus {ze Cr<< R). Construct G(z,C) and discuss the method of images or other techniques used in its derivation. 3. Solving the Dirichlet Problem: Use the Green's function derived in part 1 to solve the Dirichlet problem for a harmonic function on D with boundary condition (e) = (0), where is a continuous function on JD. 4. Poisson Integral Formula Derivation: ⚫ Derive the Poisson Integral Formula for the unit disk using Green's functions or alternatively through the method of conformal mapping and harmonic function expansion. 5. Application to Electrostatics: . Apply the Green's function for the unit disk to determine the electric potential due to a point charge inside the disk, assuming the boundary is held at zero potential. Discuss the physical interpretation of image charges in this context.
Review of complex numbers 2122=R₁₂+82) 01+02 1=R₁e 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. Ca-exi/8 R Caz-Re(6+2x/8) Explore the use of Green's functions in solving boundary value problems in complex domains. 1. Green's Function for the Unit Disk: Derive the Green's function G(, C) for the unit disk D = {C||] <1} with a pole at € D. Show that G(:,C) = log || 이는 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 |z2=z.Z= Re Re = R2e0 = R2 => |z|= √√zz=√√a²+b² = R. 2. Green's Function for Multiply Connected Domains: Extend the concept of Green's functions to a doubly connected domain D, such as an annulus {ze Cr<< R). Construct G(z,C) and discuss the method of images or other techniques used in its derivation. 3. Solving the Dirichlet Problem: Use the Green's function derived in part 1 to solve the Dirichlet problem for a harmonic function on D with boundary condition (e) = (0), where is a continuous function on JD. 4. Poisson Integral Formula Derivation: ⚫ Derive the Poisson Integral Formula for the unit disk using Green's functions or alternatively through the method of conformal mapping and harmonic function expansion. 5. Application to Electrostatics: . Apply the Green's function for the unit disk to determine the electric potential due to a point charge inside the disk, assuming the boundary is held at zero potential. Discuss the physical interpretation of image charges in this context.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 38RE
Related questions
Question
![Review of complex numbers
2122=R₁₂+82)
01+02
1=R₁e
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.
Ca-exi/8
R
Caz-Re(6+2x/8)
Explore the use of Green's functions in solving boundary value problems in complex domains.
1. Green's Function for the Unit Disk:
Derive the Green's function G(, C) for the unit disk D = {C||] <1} with a pole
at € D. Show that
G(:,C) = log ||
이는
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
|z2=z.Z= Re Re = R2e0 = R2 => |z|= √√zz=√√a²+b² = R.
2. Green's Function for Multiply Connected Domains:
Extend the concept of Green's functions to a doubly connected domain D, such as an
annulus {ze Cr<< R). Construct G(z,C) and discuss the method of images or
other techniques used in its derivation.
3. Solving the Dirichlet Problem:
Use the Green's function derived in part 1 to solve the Dirichlet problem for a harmonic
function on D with boundary condition (e) = (0), where is a continuous function
on JD.
4. Poisson Integral Formula Derivation:
⚫ Derive the Poisson Integral Formula for the unit disk using Green's functions or alternatively
through the method of conformal mapping and harmonic function expansion.
5. Application to Electrostatics:
. Apply the Green's function for the unit disk to determine the electric potential due to a
point charge inside the disk, assuming the boundary is held at zero potential. Discuss the
physical interpretation of image charges in this context.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F495f0fd0-ab38-41cc-aa1b-3e24f7167a80%2F975b5080-389b-465a-943e-99f9edda9572%2F271u55e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Review of complex numbers
2122=R₁₂+82)
01+02
1=R₁e
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.
Ca-exi/8
R
Caz-Re(6+2x/8)
Explore the use of Green's functions in solving boundary value problems in complex domains.
1. Green's Function for the Unit Disk:
Derive the Green's function G(, C) for the unit disk D = {C||] <1} with a pole
at € D. Show that
G(:,C) = log ||
이는
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
|z2=z.Z= Re Re = R2e0 = R2 => |z|= √√zz=√√a²+b² = R.
2. Green's Function for Multiply Connected Domains:
Extend the concept of Green's functions to a doubly connected domain D, such as an
annulus {ze Cr<< R). Construct G(z,C) and discuss the method of images or
other techniques used in its derivation.
3. Solving the Dirichlet Problem:
Use the Green's function derived in part 1 to solve the Dirichlet problem for a harmonic
function on D with boundary condition (e) = (0), where is a continuous function
on JD.
4. Poisson Integral Formula Derivation:
⚫ Derive the Poisson Integral Formula for the unit disk using Green's functions or alternatively
through the method of conformal mapping and harmonic function expansion.
5. Application to Electrostatics:
. Apply the Green's function for the unit disk to determine the electric potential due to a
point charge inside the disk, assuming the boundary is held at zero potential. Discuss the
physical interpretation of image charges in this context.
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