Review of complex numbers Z1=R1e01 Z₂ =R2e02 z=Rei⁹ 01+02 R Z122 R1 R2e1+82) C82=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. 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. (8=2xi/8 Consider a harmonic function : DR, where DCC is an open connected domain. 1. Harmonic Conjugates: ⚫ a. Prove that if u is harmonic on D and simply connected, then there exists a harmonic conjugate v such that fu+ iv is holomorphic on D. ⚫ b. Provide an explicit construction of given u in a specific example, such as u(x, y) = x² - y². 2. Maximum Principle for Harmonic Functions: ⚫ a. State and prove the Maximum Principle for harmonic functions, asserting that if u attains its maximum (or minimum) value in the interior of D, then u is constant. ⚫ b. Apply the Maximum Principle to show that a non-constant harmonic function cannot attain its maximum value inside a bounded domain. 3. Dirichlet Problem: ⚫a. Formulate the Dirichlet problem for finding a harmonic function on a bounded domain D with prescribed continuous boundary values. ⚫ b. Prove the existence and uniqueness of the solution to the Dirichlet problem using the method of harmonic conjugates or Perron's method. 4. Green's Function: ⚫ a. Define Green's function G(z, C) for the domain D with a pole at Ċ € D. ⚫ b. Derive the representation of a harmonic function u on D in terms of its boundary values and Green's function. Note that |z2zz Re Re-i = = R²e° = R² => |z|= √√√zz = √√√a² + b² = R.