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math185f08-hw5

# math185f08-hw5 - 4 Let S = x iy ∈ C | x,y ∈[0 1 be the...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 5 1. Consider the n th Taylor polynomial approximant to exp( z ), f n ( z ) := 1 + z + 1 2! z 2 + ··· + 1 n ! z n . Show that for all z C with Re( z ) < 0, | exp( z ) - f n ( z ) | ≤ | z | n +1 for all n N . 2. Let f : C C be an entire function. As usual, we write f ( z ) = u ( x,y ) + iv ( x,y ) for z = x + iy . (a) Show that if u is positive valued, ie. u ( x,y ) > 0 for all x,y R , then f is constant. (b) Show that if | u ( x,y ) | < | v ( x,y ) | for all x,y R , then f is constant. (c) Can we still draw the same conclusions if ‘ < ’ is replaced by ‘ > ’ in (a) and (b)? 3. Let f : C C be an entire function. (a) Show that if f satisﬁes the following conditions f ( z + 1) = f ( z ) , f ( z + i ) = f ( z ) for all z C , then f is constant. (b) Let α,β C × be such that α/β / R . Show that if f satisﬁes the following conditions f ( z + α ) = f ( z ) , f ( z + β ) = f ( z ) for all z C , then f is constant. (c) Show that if f satisﬁes the following conditions f ( z + 1) = f ( z ) , f ( z + 2) = f ( z ) for all z C , then f is constant.

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Unformatted text preview: 4. Let S = { x + iy ∈ C | x,y ∈ [0 , 1] } be the unit square in C . Let f be analytic on a region Ω that contains S . Suppose the following is true: (i) for all z with Re( z ) = 0, 0 ≤ Im( z ) ≤ 1, f ( z + 1)-f ( z ) ≥ 0; (ii) for all z with 0 ≤ Re( z ) ≤ 1, Im( z ) = 0, f ( z + i )-f ( z ) ≥ . Show that f is constant. 5. (a) Let f and g be entire functions that satisfy | f ( z ) | < | g ( z ) | for all z ∈ C . Show that there exists a constant λ ∈ C such that f ( z ) = λg ( z ) for all z ∈ C . Date : October 17, 2008 (Version 1.0); due: October 24, 2008. 1 (b) Determine all entire functions f that satisﬁes | f ( z ) | < | f ( z ) | for all z ∈ C . 2...
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math185f08-hw5 - 4 Let S = x iy ∈ C | x,y ∈[0 1 be the...

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