Review of complex numbers =Rje 22=R₂e z=Re 2122 R1 R₂+82) Caz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that 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. Ca-e2/8 |z2=zz Re Re = R2e0 = R2 => |z|= √√zz = √√a² + b² = R. Problem 5: The Cauchy Integral Formula and Its Consequences Statement: Let D be a bounded, simply connected domain in C with a piecewise smooth boundary D, and let f: D→ C be holomorphic on D and continuous on D. 1. Cauchy Integral Formula: Prove the Cauchy Integral Formula: For any z € D, 2. Cauchy's Estimates: f(C) f(z) = 2mi Using the Cauchy Integral Formula, derive Cauchy's estimates for the derivatives of f. Specifically, show that for any n≥ 0, |f(") (=)|≤ n!M R where M = maxcap |f(C) and R is the distance from z to OD. 3. Taylor and Laurent Series: Expand f into its Taylor series around a point zo € D. Prove the convergence of this series within the radius of convergence determined by the distance to the nearest singularity. ⚫ Similarly, discuss the Laurent series expansion of f in an annular region and prove its convergence.