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.