Review of complex numbers 21=R1e01 Z₂ =R2e02 z=Reia 01+02 R Z122 R1 R2e(+82) $82=Re¹(6+2x/8) The complex conjugate of z = Rei = 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. C8=2xi/8 Let f (z) be a function holomorphic in an annulus A = {ze C|r<|z-zo|< R}, where 0< T |z|= √√zz = √√√ a² + b² = R. f(z) = a(zzo)", where the coefficients an are given by and C is a positively oriented, simple closed contour within A. ⚫ b. Demonstrate the uniqueness of the Laurent series representation for f(z) in the annulus A. 2. Classification of Singularities: ⚫ a. Using the Laurent series, classify the singularity of f(z) at zo as removable, a pole, or an essential singularity based on the behavior of the coefficients an. b. Provide examples of functions exhibiting each type of singularity and illustrate their Laurent series expansions. 3. Residue Calculation: ⚫ a. Define the residue of f(z) at an isolated singularity zo and express it in terms of the Laurent series coefficients.