Complex Power Series Radius of Convergence Proof

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Homework Help Overview

The discussion revolves around a power series defined as f(z) = ∑ an(z - z0)n, with a radius of convergence R > 0. The original poster is tasked with demonstrating that if f(z) = 0 for all z within a certain radius, then all coefficients an must be zero.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to reason through the implications of the power series summing to zero, questioning whether the coefficients must also be zero or if they could potentially alternate. Some participants suggest using the Open Mapping Theorem and exploring the derivatives of f(z) to derive information about the coefficients. Others engage in a back-and-forth about the implications of taking derivatives and evaluating them at specific points.

Discussion Status

Participants are actively exploring the relationship between the function being zero and the coefficients of the power series. Some have proposed methods involving derivatives to establish connections between the coefficients and the function's behavior. There is a recognition of the need to evaluate derivatives at the correct point, which indicates a productive direction in the discussion.

Contextual Notes

There is an emphasis on the conditions under which the function f(z) equals zero and the implications for the coefficients of the series. The discussion reflects uncertainty about the correct approach and the assumptions being made regarding the behavior of the series and its derivatives.

jsi
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Homework Statement



If f(z) = [itex]\sum[/itex] an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0.

Homework Equations





The Attempt at a Solution



I know it is a power series and because R is positive I know it converges. And if f(z) = 0 then the sum itself would be 0 which would mean that each term must add to 0. This doesn't necessarliy imply that each coefficient ai is 0 though because they could alternate a1 = 1, a2 = -1, a3 = 2, a4 = -2 etc... Am I right in thinking this? And if so, I'm not sure how to go from there... Unless f(z) = 0 for all z implies that |z - z0| ≠ 0 somehow then for f(z) to equal 0, the ai would have to equal 0? Any help would be appreciated, thanks!
 
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Use Open Mapping Theorem.
 
It's pretty easy to see that all of the derivatives of f(z) are zero at z0, isn't it? Try to use derivatives to say something about the coefficients.
 
oh ok, so if say, since f(z) = 0, d/dz = 0, then taking the derivative of every term of the series yields d/dz (an(z - z0)n) = (nan(z - z0)n-1) = (nan(z - z0)n)/(z - z0)1 and then that means (z - z0) can't = 0 so the ai must be 0? Is that right, or sort of right, or totally wrong?
 
jsi said:
oh ok, so if say, since f(z) = 0, d/dz = 0, then taking the derivative of every term of the series yields d/dz (an(z - z0)n) = (nan(z - z0)n-1) = (nan(z - z0)n)/(z - z0)1 and then that means (z - z0) can't = 0 so the ai must be 0? Is that right, or sort of right, or totally wrong?

Take it one step at a time. The series is f(z)=a0+a1(z-z0)+a2(z-z0)^2+... Since f(z0)=0 that means a0=0. f'(z)=a1+2*a2(z-z0)+... What does f'(z0)=0 tell you? Now use the second derivative to say something about a2. Etc.
 
I think I got it; f'(0) = 0 implies that a1 = 0 and f''(0) = 0 implies that a2 = 0, so f to the nth derivative of 0 = 0 implies that an = 0, so all an must be 0?
 
jsi said:
I think I got it; f'(0) = 0 implies that a1 = 0 and f''(0) = 0 implies that a2 = 0, so f to the nth derivative of 0 = 0 implies that an = 0, so all an must be 0?

Yes, that's it. Except you want to evaluate all of the derivatives at z=z0. Not z=0.
 
great, thank you!
 

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