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Mathematics, 20.02.2020 21:59 studybug779
Show that a harmonic function u(x, y) is equal at every point a to its average value on any circle centered at a [and lying in the region where f (z) = u(x, y) + iv(x, y) is analytic]. Hint: In (3.9), let z = a + reiθ (that is, C is a circle with center at a), and show that the average value of f(z) on the circle is f(a) (see Chapter 7, Section 4 for discussion of the average of a function). Take real and imaginary parts of f(a) = [u(x, y) + iv(x, y)]z=a.
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