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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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- Fall '09