You may use the fact that if f z 0 for all z D then f is constant b Let f be an

You may use the fact that if f z 0 for all z d then f

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You may use the fact that if f 0 ( z ) = 0 for all z D , then f is constant. (b) Let f be an entire function and M be a positive constant ( M > 0) such that | f ( z ) | ≥ Me - y for all z = x + iy C . Show that we can find a constant k C such that f ( z ) = ke iz for all z C . (c) Establish the following integration formula with the aid of residues: Z -∞ x sin x ( x 2 + 1)( x 2 + 4) dx = π 3 ( e - 1 - e - 2 ) . Complete explanations are required. MATH2101 PLEASE TURN OVER 1
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3. (a) State Cauchy’s integral formulas for f and its derivatives. State Cauchy’s inequalities for f and its derivatives. Make sure you explain all quantities and functions that appear. (b) Let f be entire and assume that for some M > 0 we have | f ( z ) | ≤ M (1 + | z | ) 1 / 3 , z C . Show that f is a constant function. (c) Let f be holomorphic on an open set U containing the closed disc D (0 , R ). Let C be the circle centered at 0 with radius R traversed anticlockwise and let z D (0 , R ). Show that f ( z ) + f ( - z ) 2 = 1 2 πi Z C f ( w ) w w 2 - z 2 dw, f 0 ( z ) - f 0 ( - z ) 2 = 1 2 πi Z C f ( w ) 2 wz ( w 2 - z 2 ) 2 dw. 4. (a) Show that the map w = T ( z ) = z - i z + i maps the upper half-plane { z ; = ( z ) > 0 } conformally onto the unit disc { w ; | w | < 1 } . (b) Explain why the map w = T ( z ) = z - i z + i maps the first quadrant { z ; < ( z ) > 0 , = ( z ) > 0 } conformally onto the lower half-disc { w ; | w | < 1 , = ( w ) < 0 } . (c) Find a conformal map from the first quadrant { z ; < ( z ) > 0 , = ( z ) > 0 } onto the upper half-plane { z ; = ( z ) > 0 } .
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