Uniformly convergent series and products of entire functions

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SUMMARY

The discussion centers on the uniform convergence of series and products of entire functions, specifically addressing the relationship between the uniform convergence of a sequence of functions \( a_n \) and the uniform convergence of the product \( \prod(1 + a_n) \). It is established that if \( a_n(x) \) converges uniformly to \( a(x) \), then \( a_n(x) + 1 \) also converges uniformly to \( a(x) + 1 \). The participant clarifies that uniform convergence is not applicable to sequences of constants, as they do not vary with \( x \).

PREREQUISITES
  • Understanding of uniform convergence in the context of function sequences
  • Familiarity with entire functions and their properties
  • Knowledge of series and products in mathematical analysis
  • Basic concepts of limits and continuity in real analysis
NEXT STEPS
  • Study the implications of uniform convergence on the continuity of function products
  • Explore the Weierstrass M-test for uniform convergence of series
  • Investigate the properties of entire functions and their convergence behavior
  • Learn about the relationship between uniform convergence and pointwise convergence
USEFUL FOR

Mathematicians, advanced students in analysis, and anyone studying the convergence properties of entire functions and their applications in complex analysis.

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If the sum of a sequence of functions a_n converges uniformly, how is it that the product of 1+a_n converges uniformly? I know that this is true if the a_n are constants but how does this translate to functions?
 
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It seems to me almost trivial. If [math]a_n(x)[/math] converges uniformly to a(x) then:

Given any \epsilon&gt; 0, there exist N such that if n> N, [/itex]|a_n(x)- a(x)|< \epsilon[/itex] for all x.

Well, [/itex]|(a_n(x)+ 1)- (a(x)+ 1)|= |a_n(x)- a(x)|[/itex] for all x! So it immediately follows that a_n(x)+ 1 converges uniformly to a(x)+ 1.

(Oh, and it really does not make sense to talk about a sequence of constants converging "uniformly" since different x values will make no difference.)
 
I asked if \sum |a_n(z)| converges uniformly, does this imply \prod(1+a_n(z)) converges uniformly?
 

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