#19 and #18 f(z) dz Im f(z) dz? Give reason. 18. Is Im |f sa d = f.1se09| f(z) dz? f(z) dz| 19. Is f(z) dz Im f(z) dz? Give reason. 18. Is Im |f sa d = f.1se09| f(z) dz? f(z) dz| 19. Is
19. Which of the following statements are always true? (i) Re(2)Im(iz) 0 (ii) Re(iz)Im(z) = 0 (iii) z- Zi Im(z) = 0 (a) (ii) only (b) (i) only (c) (iii) only (d) (i) and (ii) only (e) (ii) and (ii) only
Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?
Find the disk of convergence of power series IM: (= –2+i)" 2" where, z = x + iy n=0
3. S how that if z is a complex number, a. Rele)-2 b. Im(2)- 2 2i 2 ίθ d. sin θ 2i
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on 3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on
2. Use the simplex algorithm to find an optimal solution to the following LP: max z 5x1 + 3x2 + x3 5x +3x2 +6x s 15
Let f(z) be entire and such that Im f(z)S cfor all z. By considering the function ee show that f(z) is a constant. -if (z)
We close E-30V max C tine im τ-units al culate Cuert reach Lmax 4) what is V t t 2?