Advanced Calculus fi..

2f 'c) J, zeZ dz on the line segment joining the endpointsd) j";~~f dz on the parabola zy2 = r + 1Chapter 8 Functions of a Complex Variable 5592. a) Evaluate Ji,(dz/z) on the path z = ei', -n/2 5 t 5 n/2, with the aid of the relation(log z)' = l/z, for an appropriate branch of log z.b) Evaluate A-'(dzlz) on the path z = elt, n/2 5 t 5 3x12, as in part (a).C) Why does the relation (log 2)' = l/z not imply that the sum of the two integrals ofparts (a) and (b) is zero?3. A certain function f (z) is known to be analytic except for z = 1, z = 2, z = 3, and it isknown thatCAwhere Ck is a circle of radiuswith center at z = k. Evaluateon each of the following paths: 4 ,,a) lzl = 4 b) lzl = 2.5 c) Iz + 2.51 = 14. A certain function f (2) is analytic except for z = 0, and it is known thatlim z f (z) = 0.z+mShow thatf (2) dz = 0ton every simple closed path not passing through the origin. [Hint: Show that the value ofthe integral on a path lzl = R can be made as small as desired by making R sufficientlylarge.]5. Evaluate each of the following with the aid of the Cauchy integral formula:a) f& dr on lzl = 5 b) f fi dz on lzl = 1e) $* dz on IZI = 2 d) f* dz on lzl = 2z +1[Hint for (c) and (d): Expand the rational function in partial fractions.]6. Prove (8.53) under the hypotheses stated.7. Prove that if f (z) is analytic in domain D and f '(2) = 0, then f (z) = constant. [Hint:Apply Theorem 14.11We now proceed to enlarge the class of specific analytic functions still further byshowing that every power seriesconverging for some values of z other than z = zo represents an analytic function.