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1. Verify the averaging property for the harmonic function u(x, y) = x 2 − y 2 at the point (x, y) = (2, 0) using a circle of radius one. 2. Consider a polar coordinates r ≥ 0 and θ ∈ [0, 2π] and consider a continuous harmonic function u(r, θ), which hence satises ∆2u = 0. Let u¯(r) = 1 2π Z 2π 0 u(r, θ)dθ Find and solve a differential equation for u¯(r), and show that u¯ is in fact a constant. (Hint: use Laplace's equation in polar coordinates as inspiration). This is another proof of the averaging property. 3. Consider a harmonic function u(x, y) dened on the infinite strip x (−∞, +∞) and y ∈ [0, L]. Let boundary data be given by u(x, 0) = 0 and u(x, L) = exp(−x 2 /2) and define uˆ(k, y) as the Fourier transform of u in the x-direction. Find uˆ(k, y). (No need to attempt to find u(x, y) explicitly.) 4. Use polar coordinates to find the solution u(r, θ) to Laplace's equation inside the unit circle that satises the following boundary conditions on the circle's circumference: u = 1 if y > 0 and u = 0 if y < 0. 5. Adapt the method from class to show that the boundary-value problem for Laplace's equation with Neumann conditions on the boundary of the domain has a unique solution up to an arbitrary constant. (Here Neumann conditions means that the normal derivative ∂u ∂n is specified on the boundary.) 6. Verify that the real and imaginary parts of the analytic function f(z) = 1 z − i are harmonic. State the averaging property for the real and imaginary part on concentric circles around the origin. What restriction do you have to put on the circle's radius for this to work?

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