7. Let z x+y (a) Show that f(z) z3 is analytic. 4 marks Recall the Caucy-Riemann equations are: ди ди an d_ where f (z)...
Tutorial Group/Date/Time: Using the Cauchy-Riemann equations, show that f(z)-e' is fully analytic in the entire z-plane. 1. (40 marks)
7. Let f:D + C be a complex variable function, write f(x) = u(x, y) +iv(x,y) where z = x +iy. (a) (9 points) (1) Present an equivalent characterization(with u and v involved) for f being analytic on D. (Just write down the theorem, you don't need to prove it.) (2) Let f(z) = (4.x2 + 5x – 4y2 + 3) +i(8xy + 5y – 1). Show that f is an entrie function. (3) For the same f as above,...
9 and 11 please 2-11 CAUCHY-RIEMANN EQUATIONS Are the following functions analytic? Use (1) or (7). 2. f(z) = izz 3. f(z) = e -2,0 (cos 2y – i sin 2y) 4. f(x) = e« (cos y – i sin y) 5. f(z) = Re (z?) – i Im (32) 6. f(x) = 1/(z – 25) 7. f(x) = i/28 8. f(z) = Arg 2TZ 9. f(z) = 3772/(23 + 4722) 10. f(x) = ln [z] + i Arg z...
pi over 2 is not correct either Let F(x, y, z) = z tan-(y2)i + z3 In(x2 + 2)j + zk. Find the flux of F across S, the part of the paraboloid x2 + y2 + z = 5 that lies above the plane z = 4 and is oriented upward.
Problem 6 (6 points) Let f(x) = u(x, y) + iv(x, y) be a analytic function on D and extends continuously to ad. Prove that the component function u(x, y) must attain its minimum value on aD unless u(x,y) is a constant function. (Hint: Consider the modulus of analytic function g(z) = ef(x), and apply the result in problem 5)
1.) Let f(x, y, z) = xy?z3 – sin(xy) +erz? – tan(y, 3. Determine the following. (a) e (b) (c) azonos
(%) = u(x, y) + f 0(4,7) For each of the following functions, write as f(z) = u(x, y) + í v(x, y) and use the Cauchy-Riemann conditions to determine whether they are analytic (and if so, in what domain) a. f(z) = 2 + 1/(2+2) b. f(z) = Re z C. f(x) = e-iz d. f(z) = ez? 16 marks]
[3](4 pts) Let f(x) = u(x, y) + iv(x,y) be differentiable for all z = x + iy. If v(x, y) = x + xy + y2 – x2, for all (x, y), find u(x,y) and express f(x) explicitly in terms of z.
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top hemisphere of the unit sphere oriented upward, and C the unit circle in the xy-plane with positive orientation. (a) Compute div(F) and curl(F). (b) Is F conservative? Briefly explain. (c) Use Stokes’ Theorem to compute ? F · dr by converting it to a surface integral. (The integral is easy if C you set it up correctly) 4. (23 pts)...