( 20 points) 2a. Determine the points for which f(z) 20 2-122 is analytic. x 1S...
6. Determine the sets on which the following functions are analytic. a. f(z) = Log(z + 1) b. f(x) = Log(x+)
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
(20 pts) Use the Cauchy-Riemann Equations to determine if the following functions are analytic or not. a) f(x)=sel.cosy Dutietsiny x3+xyz x2y+y3 b) f(2)=; +j x+y *+
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) -u(x, y) + iv(x, y). (b) Let x2 and y 1 such that z-2i is a solution to 2abi [3 marks] Determine a and b (c) Find all other solutions of 23-a + bi in polar form correct to 2 significant 3 marks] figures If you were not able to solve for a and b...
(%) = 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]
Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C. Assume f(x) = 0 for z on C. Show f(2)=0 for all z inside C.
Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey + e*, ye*). (a) (5 points) Compute V F(x, y, z). (b) (10 points) Find a potential function for F or explain why none exists. (c) (5 points) Find ScF. dr, where C is the curve consisting of the line segments from (0,0,0) to (1,2,0), from (1,2,0) to (1,2, 1), and from (1, 2, 1) to (1,2,2).
Byty 4) (20 pts) Use the Cauchy-Riemann Equations to determine if the following functions are analytic or I a) f(x) = e* (cosy + 1) + je*siny not. +
Problem 2. (15 points) a) Find the real part u(x,y) and imaginary part v(x,y) of f(z) = (1+2i)z+ (i – 1)2 +3 b) Verify if the above function is analytic c) Using Laplace's equation verify if the real part u(x,y) is harmonic.
Determine the set of points at which the function is continuous. f(x, y, z) = 7x + 2y + z D = = x, y, z) | 2 24v 7x + 2y } * Need Help? Read It Talk to a Tutor