(p). Assume that u(z) is a real harmonic function defined on the disc D(zo, p) = {lz- zolp}. Use Cauchy's integral formula to prove the mean value property: -S u(zo) = [ 2π u(zo + reio) do 27T =

Find the poles of f(z) = 1/sin(z) and determine their residues.

Verify if this is harmonic

In(x² + y2) is harmonic function and find the conjugate harmonic Q2: Show that u = v = (x. y) and from an analytic function f(z) = u(x. y) + (x. y)

Show that f(x, y) = x2 - y2 is harmonic. Verify the mean-value property for f(x, y) directly [expand f(a + r cos 0, b + r sin0) as a function of 0 and compute Icp(r)]. Show that x2 + y2 is not harmonic and does not satisfy the mean-value property.

Find the points on the graph of z = xy' + 8y¯' where the tangent plane is parallel to 9x + 5y + 9z = 0. %3D %3D (Give your answer as a comma-separated list of points in the form (*, *, *). Express numbers in exact form. Use symbolic notation and fractions where needed.) point(s):

Show whether the function is harmonic and find a harmonic conjugate v of u