this is about complex numbers Taylor series and singularity

Question 4.1: What is a singular point? What is the order of singularities? What is the Cauchy's integral theorem? What is the Cauchy's integral formula? What are the Taylor series expansion and Laurent series expansion of complex functions? What are the residues of complex functions? Question 4.2: Determine the convergence or divergence of the following series: (i) (ii) Question 4.3: Consider the function f (=) =- cosh(=) (a) Let f(=) = _a,,=" be the Taylor series expansion of f(=) around = = 0. Determine do, a1, az. (b) Let f(=)= [b. =-|be the Taylor series expansion of f(=) around = =- . Determine bo, #1 =0 b1, b2. Simplify the resulting expressions as much as possible. Question 4.4: Find the singularities of f (2) _ e - sin z -1 22 Question 4.5: Calculate - - dz . 1/ 2 23 -1 Question 4.6: Calculate 6 +dz He sin z Question 4.7: Calculate Z dz . 1=/ 2 27 + 2 2+ 2