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SOLVED:Prove that if a harmonic form is coderived, then it is identically zero.

SOLVED:Prove that if a harmonic form is coderived, then it is identically zero.

SOLVED: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) i v(x, y) is analytic]. Hint: In (3.9), let z=a r e^i θ (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) i v(x, y)]z=a

SOLVED: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) i v(x, y) is analytic]. Hint: In (3.9), let z=a r e^i θ (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) i v(x, y)]z=a

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