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Question 5. For e > 0, define U₂ = {z E C: Re(z) > 1 + }, and for integers n ≥ 1, define (n(z) = n*, where the principal branch of the logarithm is taken in the definition of the complex power. (a) Show that Sn (2)| ≤n-(1+e) for all n and z € U₂.

Question 5. For e > 0, define U₂ = {z E C: Re(z) > 1 + }, and for integers n ≥ 1, define (n(z) = n*, where the principal branch of the logarithm is taken in the definition of the complex power. (a) Show that Sn (2)| ≤n-(1+e) for all n and z € U₂.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 5. For e > 0, define U₂ = { z = C: Re(z) > 1 + ε}, and for integers n ≥ 1, define (n(z) = n*, where the principal branch of the logarithm is taken in the definition of the complex power. (a) Show that n (2)| ≤ n−(¹+) for all n and z € U₂. (b) Deduce that En-1 Sn (2) converges uniformly on U₂. (c) Deduce that the function defined by ς(2) = Σ Sn(2) = Σ = n=1 n=1 is holomorphic on { z = C: Re(z) > 1}. nz 100 n=1 For part (b) you may assume that the series -1 converges if p > 1. This function is the celebrated Riemann zeta function. In fact this function can be extended to be defined on C\{1}. As an interesting exercise, you might like to prove that if z = C\{1} satisfies (z) = 0, then either z = -2k for a positive integer k, or Re(z) = 1/2, but please don't spend too long trying.
Transcribed Image Text:Question 5. For e > 0, define U₂ = { z = C: Re(z) > 1 + ε}, and for integers n ≥ 1, define (n(z) = n*, where the principal branch of the logarithm is taken in the definition of the complex power. (a) Show that n (2)| ≤ n−(¹+) for all n and z € U₂. (b) Deduce that En-1 Sn (2) converges uniformly on U₂. (c) Deduce that the function defined by ς(2) = Σ Sn(2) = Σ = n=1 n=1 is holomorphic on { z = C: Re(z) > 1}. nz 100 n=1 For part (b) you may assume that the series -1 converges if p > 1. This function is the celebrated Riemann zeta function. In fact this function can be extended to be defined on C\{1}. As an interesting exercise, you might like to prove that if z = C\{1} satisfies (z) = 0, then either z = -2k for a positive integer k, or Re(z) = 1/2, but please don't spend too long trying.
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