A function g(x, y) is said to be harmonic at a point (x0, y0) if g is of class C2 and

Chapter 6, Problem 31

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A function g(x, y) is said to be harmonic at a point (x0, y0) if g is of class C2 and satisfies Laplaces equation 2g = 2g x 2 + 2g y2 = 0 on some neighborhood of (x0, y0). We say that g is harmonic on a closed region D R2 if it is harmonic at all interior points of D (i.e., not necessarily along D). Exercises 3033 concern some elementary results about harmonic functions in R2.Let f be harmonic on a region D that satisfies the assumptions of Exercise 30. Show that D f f dA = D f f n ds