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. Z2=R2e2 21=R1e01 z=Reia (8=e²xi/8 01+02 R R Z122 R1 R2e1+82) $82=Rei(+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-i = R²e = R² => |z|= √√√zz = √√√a² + b² = R. Let D and D' be simply connected domains in C, and let f : DD' be a biholomorphic (conformal) map. 1. Preservation of Laplace's Equation: Show that if u : D'→R is a harmonic function on D', then u of is a harmonic function on D. 2. Conformal Invariance of the Laplacian: • Prove that the Laplacian operator is conformally invariant up to a scaling factor. Specifically, demonstrate that for f: DD' biholomorphic and u twice continuously differentiable, A(uf) = f(z)|(Au) of. 3. Dirichlet Problem and Conformal Mapping: ⚫ Consider the Dirichlet problem for a harmonic function on D' with boundary values given by a continuous function : OD'→R. Use the conformal map f to transform this problem to D and solve for the harmonic function on D. 4. Schwarz-Pick Lemma Extension: • Extend the Schwarz-Pick Lemma to bounded symmetric domains and discuss its implications for the hyperbolic metric under biholomorphic mappings. 5. Applications to Fluid Dynamics: Apply the results from parts 1-3 to model potential flow around a simple obstacle. Specifically, use a conformal map to transform the flow domain and solve for the velocity potential and stream function.