Homework Answers
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 harmoni...
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,...
. (a) Show that the function u= 4x2 - 12.xy2 is harmonic and v=12.xy-4v2 is a...
. (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 .
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its...
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...
S. (20 points) Show that u cos is harmonic and find its harmonic conjugate v y...
S. (20 points) Show that u cos is harmonic and find its harmonic conjugate v y ).
Let A = ∂ 2w/∂x2 , B = ∂ 2w/∂x∂y, C = ∂ 2w/∂y2 . From...
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.)
Solve the separable initial value problem. tan(sin(x^(2) 1. y' = 2x cos(x2)(1 + y2), y(0) =...
Solve the separable initial value problem. tan(sin(x^(2) 1. y' = 2x cos(x2)(1 + y2), y(0) = 5 → y= 2. v' = 8e4x(1 + y2), y(0) = 2 + y=
Prove that u (x, y) is harmonic and find its conjugate harmonica (v (x, y)). Additionally...
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
differential equations 2. (a) Verify that yı = e cos x and y2 = etsin x...
differential equations
2. (a) Verify that yı = e cos x and y2 = etsin x are solutions of -2y + 2yło on (-00,00). 204 Chapter 5 Linear Second Order Equations (b) Verify that ifc, and are arbitrary constants then y = cre* cos x + cze sinx is a solution of (A) on (-00,00) (c) Solve the initial value problem y" - 2y + 2y = 0, y(0) = 3. y'(O) = -2
(1 point) Show that the function f(x, y) = ux4 – 2x”y – 18x²y2 + vxy3...
(1 point) Show that the function f(x, y) = ux4 – 2x”y – 18x²y2 + vxy3 + wyt is harmonic, i.e. satisfies Laplace's equation af + əx2 a2f dy2 0 if and only if the constants U, V, w are given by U = V = W =
Q7 Prove the real valued function in x and y given by 1) and (ii) are...
Q7 Prove the real valued function in x and y given by 1) and (ii) are harmonic. Find the corresponding harmonic conjugate function and hence construct the analytic function f(z) = u(x,y) +j v (x,y) 0v(x, y) = In(y2 + x2) + x + y, z = 0 (ii) u(x,y) = y2 – x2 + 16xy
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,...
. (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 .
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...
S. (20 points) Show that u cos is harmonic and find its harmonic conjugate v y ).
Solve the separable initial value problem. tan(sin(x^(2) 1. y' = 2x cos(x2)(1 + y2), y(0) = 5 → y= 2. v' = 8e4x(1 + y2), y(0) = 2 + 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
differential equations
2. (a) Verify that yı = e cos x and y2 = etsin x are solutions of -2y + 2yło on (-00,00). 204 Chapter 5 Linear Second Order Equations (b) Verify that ifc, and are arbitrary constants then y = cre* cos x + cze sinx is a solution of (A) on (-00,00) (c) Solve the initial value problem y" - 2y + 2y = 0, y(0) = 3. y'(O) = -2
(1 point) Show that the function f(x, y) = ux4 – 2x”y – 18x²y2 + vxy3 + wyt is harmonic, i.e. satisfies Laplace's equation af + əx2 a2f dy2 0 if and only if the constants U, V, w are given by U = V = W =
Q7 Prove the real valued function in x and y given by 1) and (ii) are harmonic. Find the corresponding harmonic conjugate function and hence construct the analytic function f(z) = u(x,y) +j v (x,y) 0v(x, y) = In(y2 + x2) + x + y, z = 0 (ii) u(x,y) = y2 – x2 + 16xy
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