118Homework3

118Homework3 - eralize to complex functions in an often...

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Homework 3 Due Monday, Sept. 13 1. Recall that a function h : R 2 R is harmonic if it satisfies Laplace’s equation: h xx + h yy = 0. Suppose that f : C C is differentiable with f ( z ) = u ( x, y )+ iv ( x, y ). Prove that if h : R 2 R is a harmonic function, then the function h ( u ( x, y ) , v ( x, y )) is also harmonic. (You may assume that the second partial derivatives exist and are continuous.) 2. Prove that if v is a harmonic conjugate of u , then - u is a harmonic conjugate of v . 3. Prove that if v is a harmonic conjugate of u , and if u is a harmonic conjugate of v , then u and v are constant. 4. Suppose that A , B , and C are constants and consider the quadratic h ( x, y ) = Ax 2 + Bxy + Cy 2 . (a) What are necessary and sufficient conditions on A , B , and C for h ( x, y ) to be harmonic? (b) For such an A , B , and C as in Part (a), find all harmonic conjugates of h ( x, y ). 5. We have seen that many results for derivatives of real-valued functions gen-
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Unformatted text preview: eralize to complex functions in an often straightforward manner. However, there are a few important results that do not generalize. Recall the Mean Value Theorem for real-valued functions: Mean Value Theorem: If f : R R is continuous on [ a, b ] and dierentiable on ( a, b ) , then there is a point c ( a, b ) at which f ( b )-f ( a ) = f ( c )( b-a ) . Show that the Mean Value Theorem does not hold for complex valued functions. Hint: Let f ( z ) = e iz , let z 1 = 0, and let z 2 = 2 . Show that there is no w C with the property that f ( z 1 )-f ( z 2 ) = f ( w )( z 1-z 2 )....
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