Practice Problem 3, (2 x 8 = 16 points (a) Whether is f(a)-Re() differentiable? Explain why...
(%) = 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 1. (13 points) 1. What is the maximum modulus principle? (3 pts) 2. Cite the Cauchy-Riemann theorem. (3 pts) 3. Give the definition of a harmonic function defined on an open subset ACR. (3 pts) 4. Prove that the real and imaginary part of a complex analytic function is harmonic. (4 pts)
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...
3. (a) (3 points) Write the definition of the derivative of a differentiable function f(x) at = a; (b) (7 points) using the definition of derivative as in (a), find the derivative of the function f(x) = Vx at a = 2. (c) EXTRA CREDIT (2 points): State the MEAN VALUE THEOREM (you can also draw a picture) and give its PHYSICAL interpretation in terms of INSTANTANEOUS and AV- ERAGE VELOCITIES.
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
3. (a) (3 points) Write the definition of the derivative of a differentiable function f() at r = a; (b) (7 points) using the definition of derivative as in (a), find the derivative of the function f(x) = at a = 2.
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
9. For each of the following, provide a suitable example, or else explain why no such example exists. [2 marks each]. a) A function f : C+C that is differentiable only on the line y = x. b) A function f :C+C that is analytic only on the line y = x. c) A non-constant, bounded, analytic function f with domain A = {z | Re(z) > 0} (i.e., the right half-plane). d) A Möbius transformation mapping the real axis...
For the function f(x), determine whether or not f is continuous and/or differentiable at the following points. Also using only the given function (not a graph), determine what occurs graphically at these points. f(x) = 1, X, x² - 12, x < 0 0<x< 4 X > 4 (a) At x = 0, f(x) is ---Select--- . At this point, the graph of f(x) has ---Select--- (b) At x = 2, f(x) is ---Select--- . At this point, the graph...
Explain why the function is differentiable at the given point. ROX. y) 6 + x In xy - 9). (5,2) The partial derivatives are x,y) and f(x, y) = .00 1,(5,2) - c and 15, 2) = Bothf, and fare continuous functions for xy > and is differentiable at (5,2) Find the linearization ( ) of (x, y) at (5,2).