**Unformatted text preview: **| ≤ 1 } except for a simple pole at z = i/ 2. If f also satisﬁes f ( 1 2 ) = 0 as well as | f ( z ) | ≤ 1 on | z | = 1, show that | f (0) | ≤ 1. B–2. Find a conformal map f ( z ) = u + iv from the unit disk {| z | < 1 } to the ﬁrst quadrant, { u > , v > } . B–3. Give a complete clear proof of the fundamental theorem of algebra: Every nontrivial complex polynomial has at least one root. B–4. Evaluate Z ∞ cos x 1 + x 2 dx . B–5. Let g ( z ) be holomorphic in the closed unit disk D = {| z | ≤ 1 } and assume that | g ( z ) | ≤ 2 for | z | = 1. How many roots does h ( z ) := g ( z ) + 5 z 3-2 have in D ? As usual, justify your assertions. B–6. Let h ( z ) = ∞ X n =1 a n n z , where z = x + iy , and assume the sequence a n is bounded, say | a n | ≤ M . Show that h ( z ) is holomorphic in the half-plane { x > 1 } ....

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- Fall '09
- Math, Fundamental Theorem Of Algebra, Taylor Series, Complex number, Conformal map, Jerry L. Kazdan