Let u(x, y) = x^3 + kxy^2 + y. (a) Determine the value of k such...

Let u(x, y) = x^3 + kxy^2 + y. (a) Determine the value of k such that u is an harmonic function. (b) Find the harmonic conjugate v of u. (c) Obtain the expression of f = u + iv in terms of z = x + iy

Expert Solution

Colby Messinger answered 1 year ago

Solve x(y^2+U)Ux -y(x^2+U)Uy =(x^2-y^2)U, U(x,-x)=1

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡...

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡ ( x ^2 + y ^3 ). Determine the line integral of f ( x , y ) with respect to arc length over the unit circle centered at the origin (0, 0).

Let f(x, y) = x^ 2 + kxy + 4y^ 2 , k a constant. The...

Let f(x, y) = x^ 2 + kxy + 4y^ 2 , k a constant. The point (0, 0) is a stationary point of f. For what values of k will f have a local minimum at (0, 0)? (a) |k| > 4 (b) k ≥ −4 (c) k ≤ 4 (d) |k| < 4 (e) none of the above The point P : (2, 2) is a stationary point of the function f(x, y) = 6xy − x ^3...

Given the utility function U ( X , Y ) = X 1 3 Y 2...

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Consider the series X∞ k=3 √ k/ (k − 1)^3/2 . (a) Determine whether or not...

Consider the series X∞ k=3 √ k/ (k − 1)^3/2 . (a) Determine whether or not the series converges or diverges. Show all your work! (b) Essay part. Which tests can be applied to determine the convergence or divergence of the above series. For each test explain in your own words why and how it can be applied, or why it cannot be applied. (i) (2 points) Divergence Test (ii) Limit Comparison test to X∞ k=2 1/k . (iii) Direct...

Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give...

Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give an explict solution for k=-2.5

Let X and Y have the following joint density function f(x,y)=k(1-y) , 0≤x≤y≤1. Find the value...

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Let u(x, y) be the harmonic function in the unit disk with the boundary values u(x,...

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Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and 0<=y<=1. a) Find k. b)...

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and 0<=y<=1. a) Find k. b) Find the joint cumulative density function of (X,Y) c) Find the marginal pdf of X and Y. d) Find Pr[Y<X2] and Pr[X+Y>0.5]

a) U = xy b) U = (xy)^1/3 c) U = min(x,y/2) d) U = 2x...

a) U = xy b) U = (xy)^1/3 c) U = min(x,y/2) d) U = 2x + 3y e) U = x^2 y^2 + xy 4. All functions except c) are differentiable. Do these functions exhibit diminishing marginal utility? Are their Marshallian demands downward sloping? What can you infer about the necessity of diminishing marginal utility for downward- sloping demands?

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