Uniformly convergent series and products of entire functions

In summary, if the sum of a sequence of functions a_n converges uniformly, then the product of 1+a_n also converges uniformly. This is true even if the a_n are functions and not just constants. This follows directly from the definition of uniform convergence and the fact that adding 1 to a function does not affect its convergence. It is not logical to talk about a sequence of constants converging uniformly since the value of x does not affect the convergence.
  • #1
ForMyThunder
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If the sum of a sequence of functions [tex]a_n[/tex] converges uniformly, how is it that the product of [tex]1+a_n[/tex] converges uniformly? I know that this is true if the [tex]a_n[/tex] are constants but how does this translate to functions?
 
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  • #2
It seems to me almost trivial. If \(\displaystyle a_n(x)\) converges uniformly to a(x) then:

Given any [itex]\epsilon> 0[/itex], 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 [itex]a_n(x)+ 1[/itex] 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.)
 
  • #3
I asked if [tex]\sum |a_n(z)|[/tex] converges uniformly, does this imply [tex]\prod(1+a_n(z))[/tex] converges uniformly?
 

1. What is the definition of a uniformly convergent series of entire functions?

A uniformly convergent series of entire functions is a series in which the sequence of partial sums converges uniformly on every compact subset of the complex plane.

2. Can a uniformly convergent series of entire functions diverge for some points in the complex plane?

Yes, a uniformly convergent series of entire functions can still diverge for some points in the complex plane. However, the divergence will be limited to a set of isolated points and the series will still converge uniformly on compact subsets of the complex plane.

3. How can I determine if a product of entire functions is uniformly convergent?

A product of entire functions is uniformly convergent if and only if the sequence of partial products converges uniformly on every compact subset of the complex plane.

4. Can a uniformly convergent series of entire functions have a finite radius of convergence?

No, a uniformly convergent series of entire functions must have an infinite radius of convergence. This is because entire functions are analytic on the entire complex plane, so the series must have a radius of convergence of infinity to ensure convergence on all points.

5. Are there any conditions for a series of entire functions to be uniformly convergent?

Yes, there are several conditions that can ensure a series of entire functions is uniformly convergent. Some of these conditions include using the Weierstrass M-test, using Cauchy's root test, and requiring the terms of the series to be bounded by a convergent geometric series.

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