Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
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,...
pls solve these three parts of the question
a) Show that the vector + =(x+2y+4z)i +(2x-3y-z)j +(4x-y-22). is irrotational and find its 7 scalar potential. b) Find the directional derivative of xyz + xz at (1, 1, 1) in a direction of the normal to the 171 surface 3xy? + y= z at (0, 1, 1). c) Find the angle between the normal to the surface x2 = yz at the points (1, 1, 1) and (2, 4, 1). (6)
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π,
4[10 pts]. Let f(z) = u (r,0)...
. (a) Show that the function u= 4x2 - 12.xy2 is harmonic and v=12.xy-4v2 is a harmonic conjugate of u. [Consequently, the function f =u+iv is entire, thus it has an antiderivative and that any contour integral of f is path independent.] (b) Find an antiderivative F(-)= F(x+iy)=P(x, y)+i Q(x, y) of the function f; and (c) evaluate ( f (2) ds , where C is any contour from 0 to 1–2i .
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)
write c++ to solve this linear system.
Given 2x+4y-z 2 -2x -3y +7z 10 Write a program to solve the linear system above using Gauss- Elimination. Question2 Let f(x) = x3-6x2 + 11x-6, Write a program to i) Check whether the root exist in the interval: a) .5, 5.0] b) 2.5, 4.0] ii Based on interval found in 4(i), find the root using Bisection method. Do your calculation in 4 decimal points and 0.0001. Programming language: C or C++
Given...
differential equations
(c) Let u = Re e +52+3+1. Show that u is harmonic function and find the harmonic conjugate v of u. [3]
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.
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...