Review of complex numbers 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. C8=e²xi/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(z, C) for the unit disk D = { € C ||| < 1} with a pole at Є D. Show that 21=R1e01 22= =R₂e z=Rei⁹ 01+02 R R Z122 R1 R2e1+82) $82=Re¹(6+2x/8) The complex conjugate of z = Reie = a + bi is z=Rea- bi, which is the reflection of z across the real axis. Note that z2z-Z Re Re R²=R² = = =>> z=√√√zz=√√√a²+b²=R. G(z, () = log Z 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 { € C❘r<|z| < 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 u on ID with boundary condition u(e) = (6), where is a continuous function on ǝD. 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 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. C8=e²xi/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(z, C) for the unit disk D = { € C ||| < 1} with a pole at Є D. Show that 21=R1e01 22= =R₂e z=Rei⁹ 01+02 R R Z122 R1 R2e1+82) $82=Re¹(6+2x/8) The complex conjugate of z = Reie = a + bi is z=Rea- bi, which is the reflection of z across the real axis. Note that z2z-Z Re Re R²=R² = = =>> z=√√√zz=√√√a²+b²=R. G(z, () = log Z 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 { € C❘r<|z| < 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 u on ID with boundary condition u(e) = (6), where is a continuous function on ǝD. 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
Section3.6: Quadratic Functions
Problem 28E
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Transcribed Image Text:Review of complex numbers
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.
C8=e²xi/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(z, C) for the unit disk D = { € C ||| < 1} with a pole
at Є D. Show that
21=R1e01
22=
=R₂e
z=Rei⁹
01+02
R
R
Z122 R1 R2e1+82)
$82=Re¹(6+2x/8)
The complex conjugate of z = Reie = a + bi is
z=Rea- bi,
which is the reflection of z across the real axis.
Note that
z2z-Z Re Re R²=R²
=
=
=>>
z=√√√zz=√√√a²+b²=R.
G(z, () = log
Z
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 { € C❘r<|z| < 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 u on ID with boundary condition u(e) = (6), where is a continuous function
on ǝD.
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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