I don't understand this question at all. I've asked it a few times, and have gotten what look like a correct solutions, but I don't understand the reasoning behind many of the steps. I'm asking it again, hoping to get an even more thorough explanation. I'm sorry, it just isn't making sense to me. Thanks for your time. 9. Let be a bounded open subset of C, and 4:Prove that if there exists a point zo EN such that2 a holomorphic function.4(20) = 20 and ' (zo) = 1then is linear.[Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near0, and prove that if yk = o...o (where y appears k times), then k(z) =z+kanzn +0(z+¹). Apply the Cauchy inequalities and let k → ∞ to concludethe proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]

9. Let 2 be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo E such that 4(20) <= 20 and (20) = 1 then 4 [Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near 0, and prove that if k = 90.0 (where appears k times), then k(z) = z+kanz" +0(z+1). Apply the Cauchy inequalities and let k→ ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.] is linear.

9. Let 2 be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo E such that 4(20) <= 20 and (20) = 1 then 4 [Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near 0, and prove that if k = 90.0 (where appears k times), then k(z) = z+kanz" +0(z+1). Apply the Cauchy inequalities and let k→ ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.] is linear.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 11EQ

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9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo EN such that 2 a holomorphic function. 4(20) = 20 and ' (zo) = 1 then is linear. [Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near 0, and prove that if yk = o...o (where y appears k times), then k(z) = z+kanzn +0(z+¹). Apply the Cauchy inequalities and let k → ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]
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