Question: ENGR3300 Advance Engineering Mathematics Assignment 4 . Exercise 1 Consider the function f ( z ) z 2 w . Graph the mapping in the

ENGR3300 Advance Engineering Mathematics Assignment 4 . Exercise 1 Consider the function f ( z ) z 2 w . Graph the mapping in the w-plane of the semicircle in the zplane shown in the figure. z-plane Exercise 2 Evaluate C z dz z 0 to z 4 2 i along the curve C given by (a) z t 2 i t (b) the line from z 0 to z 2 i and then the line from z 2i to z 4 2 i . Plot the curves C for both cases. Exercise 3 Prove that (a) dz 0 , (b) C C z dz 0 , (c) C ( z z0 ) dz 0 where C is any simple closed curve and z 0 is a constant. Hint: Utilize the Cauchy's theorem. Please note that in order to apply the theorem you have to prove the function is analytic. Exercise 4 Utilizing the Cauchy's Theorem or the Cauchy's integral formula evaluate the integrals of (a) C sin z dz where C is | z | 1 . Plot the curve C and the singularity. 2z sin z dz where C is | z | 2 . Plot the curve C and the singularity. 2z sin(2 z ) dz where C is | z | 3 . Plot the curve C and the singularity. (c) C 6z z2 dz where C is | z 2 | 1 . Plot the curve C and the singularity. (d) C z 3 (b) C (e) C z2 dz where C is . Plot the curve C and the singularity. z3 1

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