S. (20 points) Show that u cos is harmonic and find its harmonic conjugate v y...
Prove that u (x, y) is harmonic and find its conjugate harmonica (v (x, y)). Additionally graph both functions for different integration constants: 1)ular,y) = 2x(1 - y) 2)u(x,y) = 2.r - 3 + 3.xy? 3)(x, y) = sinhrsiny 4)u(x, y) = 72+y2
Let W(x, y) be a harmonic function, and also let u(x, y) and v(x, y) be a harmonic conjugate pair. Show by hand that the composite function W(u(x, y), v(x, y)) is also harmonic. Let W(x, y) be a harmonic function, and also let u(x, y) and v(x, y) be a harmonic conjugate pair. Show by hand that the composite function W(u(x, y), v(x, y)) is also harmonic.
5. Verify that the given function u is harmonic and find v, the conjugate harmonic function of u.
solve number 3 2. Show that u(x, y) = e = cos 12 + y2 is harmonic on the punctured complex plane D=C\{0}, and find a harmonic conjugate v of it. 1 2 3. Reveal, what is f(x) = u(x,y) + iv(I,y) in Problem 2?
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,...
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...
. (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 .
(a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to the upper half plane c)Find the Dirichlet Green function for the positive quadrant + (a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to...
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]
Let u = 2.c" + 2x - 2y + 3y. (a) (6 points) Show that u is harmonic (b) (10 points) Find a harmonic conjugate of u. (C) (9 points) Find an analytic function f(x) of z = x + iy such that Ref = u.