Please explain thoroughly:

4. (9 points) ASection 1.3 Indicate whether each of the statements below is true or false. If a statement is true, prove it; if a statement is false, give a counterexample. a. Arg 2122 = Argzi + Argz2, 21, 22 0. b. arg z = - arg z, if = is not a real number. c. arg(21/22) = arg z - arg 22, 21, 22 / 0. 5. (8 points) ASection 1.4 a. Use mathematical induction to prove that for any n > 1, 2n+1 _ b. Use part (a) and De Moivre's formula to establish the following identity sin ( n2 + ; ) 0 1 + cos # + cos(20) + . . . + cos(ne) = = 2 sin 6. (16 points) AnSection 1.4 a. Where is the function f(2) = ex-[cos(2ry) - isin(2ry)] differentiable? b. Write f(=) in terms of = (no I, y should appear) and calculate the derivative at all point(s) where it is differentiable. C. Let g(2) = eth = er-y cos(2ry) + i[er- sin(2ry)] (it is optional to verify this). Calculate ly and vr, where g(2) = u + iv. Note that g(2) is an entire function from the expression g(=) = e-. d. Use g'(2) = ux + ivy to evaluate g'(i). 7. (4 points) 0Sections 2.3-2.4 Let f(2) = ear + ie . Find all values of ~ where the function is differentiable. 8. (10 points) Sections 2.1-2.3 a. Let f(2) = (23 - 3i2 + wi)-2081, calculate f'(2) at all points where it is continuous (no need to find the domain) and then calculate lim f'(=). b. Let f(=) = (22 + 1)2 (23 + iz + 1)3: calculate lim f(2) and then calculate f'(2) at all points where it is differentiable