(%) = u(x, y) + f 0(4,7) For each of the following functions, write as f(z)...
7. Show that the following functions u(x, y) monic functions v(x, y) and determine f(z) = u(x,y) + iv(x, y) are harmonic, find their conjugate har- as functions of 2. 2x2 2лу — 5х — 22. Зл? — 8ху — Зу? + 2у, (а) и(х, у) (b) и(х, у) (с) и(х, у) (d) u(a, y) 2e cos y 3e" sin y, = 3e-* cos y + 5e-" sin y, = elx cos y - e y sin y, (e) u(x,...
Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D C C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not...
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...
2- a) The real part of a complex function f(z) given as, u(x, y) = 3x?y - y. Iff(2) is an analytic function, find v(x,y) and f(z) (15p) b) Find the whether f(z) is analytic or not where f(z) = cos(x) +ie'sinx. (15p)
Practice Problem 3, (2 x 8 = 16 points (a) Whether is f(a)-Re() differentiable? Explain why (b) Check whether f()-Re(2)-Im(2) is an analytic function by checking the Cauchy- Riemann equations.
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...
(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 *+
Consider the complex functions given below: a) f(z) z,(z # 0) b) f(z)1, (0) c) f()22 d)f(2)1/(z+1), (z ) Verify that the Cauchy-Riemann equations are satisfied, and evaluate f (z) expression using the basic definition of derivative operation based on the limiting case as lim Az-0 Consider the complex functions given below: a) f(z) z,(z # 0) b) f(z)1, (0) c) f()22 d)f(2)1/(z+1), (z ) Verify that the Cauchy-Riemann equations are satisfied, and evaluate f (z) expression using the basic...
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,...
Let A = ∂ 2w/∂x2 , B = ∂ 2w/∂x∂y, C = ∂ 2w/∂y2 . From the calculus of functions of two variables, w(x, y), we have a saddle point if B 2 − AC > 0. With f (z) = u(x, y) + iv(x, y), apply the Cauchy–Riemann conditions and show that neither u(x, y) nor v(x, y) has a maximum or a minimum in a finite region of the complex plane. (See also Section 7.3.)