Review of complex numbers 2=R₂e 1=R₁₂+8₂) 01+02 =Rje 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. R Csz-Ro(+2/8) The complex conjugate of z = Reie = a + bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that Iz zz Re Re = R2e0 = R2 |2|=√27=√√√2+1²=R. Let D be the upper half-plane H = {C | 3(z) > 0}, and let D' be a polygonal region in C with vertices w₁, W2,..., in counterclockwise order. 1. Schwarz-Christoffel Mapping: Prove the existence of a conformal mapping f: HD' that maps the real axis R onto the boundary of D', sending specific points (k) CR to the vertices (w). 2. Determination of Pre-Vertices: Explain how the pre-images {x} (pre-vertices) on IR are determined given the angles at the vertices {w} of D'. Discuss the role of the angle condition in the Schwarz-Christoffel formula. 3. Schwarz-Christoffel Integral: Derive the Schwarz-Christoffel integral representation for f(=), given by f(z) =A[(-) dt +B, where a are the interior angles at the vertices w, and A, B are constants. 4. Numerical Computation: ⚫Discuss the challenges involved in numerically computing the Schwarz-Christoffel mapping for a given polygonal domain D'. Outline the steps of a numerical algorithm to approximate the pre-vertices {x} and evaluate the integral.