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 diﬀerentiable 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 π . 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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This note was uploaded on 02/21/2012 for the course MATH 118 taught by Professor Forde during the Fall '09 term at University of Houston.
- Fall '09