Prove that the power series for e^z does not converge uniformly on C.

Since the power series for e^z does not converge uniformly on C, the sequence of partial sums does not converge uniformly on C. Therefore, by the proposition, the f_n are not eventually constant, which means the power series for e^z does not converge uniformly on C. In summary, we can prove that the power series for e^z does not converge uniformly on C by showing that the sequence of partial sums does not converge uniformly and using the proposition that states if a sequence of entire functions converges uniformly to 0 on C, then the functions must be eventually constant.
  • #1
michael.wes
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Homework Statement



Prove that the power series for e^z does not converge uniformly on C.


Homework Equations



[tex]e^z=\sum_{k=0}^\infty z^k/k![/tex]

The Attempt at a Solution



The hint in the problem is to prove a proposition first:

If f_n is a sequence of entire functions that converges to 0 uniformly on C, then the f_n are eventually constant.

I proved this (an easy application of uniform convergence and Liouville's theorem), but I don't see how to use this to prove the main question.

Thanks for your help!
MW
 
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  • #2
Take the f_n to be the partial sums of the power series of e^z.
 

1. Why does the power series for e^z not converge uniformly on C?

The power series for e^z does not converge uniformly on C because it fails the Cauchy criterion for uniform convergence. This means that for any positive real number r, there exists a point z in the complex plane such that the difference between the partial sum of the series at z and the actual value of e^z is greater than r.

2. Can you provide an example of a point z in the complex plane where the power series for e^z fails to converge uniformly?

One example is the point z = i. The partial sum of the series at this point is 1 + i, while the actual value of e^i is approximately 0.54 + 0.84i, which is significantly different. This shows that the series does not converge uniformly at z = i.

3. What is the Cauchy criterion for uniform convergence?

The Cauchy criterion states that a series of functions converges uniformly on a set S if and only if for any positive real number r, there exists a point in S such that the difference between the partial sum of the series at that point and the actual value of the function at that point is less than r for all subsequent terms in the series.

4. How does the failure of uniform convergence for the power series of e^z affect its analyticity?

The failure of uniform convergence does not affect the analyticity of e^z. The function e^z is still analytic on the entire complex plane, but the power series does not converge uniformly on the entire plane. This means that e^z cannot be represented by its power series on the entire complex plane, but it can still be represented by its power series on smaller subsets where uniform convergence is achieved.

5. Is there a way to modify the power series for e^z to make it converge uniformly on C?

Yes, the power series can be modified by adding a correction term to each term in the series. This is known as the Weierstrass M-test, which allows for the determination of a smaller interval on which the power series converges uniformly. However, this modified series may not represent the function e^z exactly on the entire complex plane, but it will be a good approximation on the smaller interval where uniform convergence is achieved.

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