hw14-201141 - (2 Compute Z ∞ x 2 x 4 x 2 1 dx(3 Compute Z...

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Homework 14. (1) (Schwarz’s lemma) Let f be a holomorphic function on B (0 , 1) with | f ( z ) | ≤ 1 for all | z | < 1 and f (0) = 0. a. Define f 1 ( z ) = f ( z ) z for z 6 = 0 in B (0 , 1). Prove that z = 0 is a removable singularity of f 1 . b. Prove that | f 1 ( z ) | ≤ 1 r on B (0 ,r ) for all 0 < r < 1. (Hint: use the maximum modulus principle.) c. Conclude that | f ( z ) | ≤ | z | for all z B (0 , 1). Moreover if equality holds for some z 0 6 = 0, then there exists c with | c | = 1 such that f ( z ) = cz for all z B (0 , 1).
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Unformatted text preview: (2) Compute Z ∞ x 2 x 4 + x 2 + 1 dx. (3) Compute Z ∞-∞ cos πx x 2-2 x + 2 dx by integrating f ( z ) = e πiz z 2-2 z +2 over a semi-circular path. (4) (Quals ’06) Let f be a holomorphic function on | z | < 1. Assume f ( 1 n ) ∈ R for n ≥ 2. Prove f ( x ) ∈ R for all-1 < x < 1. (Hint: Use Problem 4 from HW 8.) 1...
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