Uniqueness theorem for power series

In summary, the conversation discusses the process of solving part b) of Theorem 3.2 in Lang's Complex Analysis. The person initially struggled with the problem due to focusing on only one part of the theorem statement, but eventually realizes that part b) can be solved using the contrapositive of part a). This leads to the conclusion that h is a constant function with a value of 0.
  • #1
snipez90
1,101
5
Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZ...complex analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis.

But I think part of the reason was that I kept concentrating on the second sentence of the theorem statement in part a), instead of the entire statement. Just to make sure, part b) follows by using the contrapositive of part a) so that h reduces to a constant which is in fact 0, correct? Thanks.
 
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  • #2
Yep. Since h(x)=0 for all x in an infinite set with 0 as accumulation point, the second sentence of part (a) applied to h is NOT true. Hence (by contraposition) the first sentence of part (a) is not true, meaning h is constant. Finally, h being constant and h(x)=0 for some x, it follows that h is everywhere 0.
 

FAQ: Uniqueness theorem for power series

What is the uniqueness theorem for power series?

The uniqueness theorem for power series states that if two power series have the same coefficients and converge to the same function within a certain interval, then they must be identical for all values within that interval.

What is the significance of the uniqueness theorem for power series?

The uniqueness theorem for power series is significant because it allows us to confidently use power series to represent functions, knowing that there is only one unique representation for a given function within a certain interval.

Can the uniqueness theorem for power series be extended to multivariable functions?

Yes, the uniqueness theorem for power series can be extended to multivariable functions by using multivariable power series representations and considering the convergence within a certain region rather than an interval.

What are the limitations of the uniqueness theorem for power series?

The uniqueness theorem for power series only applies within a specific interval or region, and it does not guarantee convergence outside of that interval or region. Additionally, there may be functions that cannot be represented by power series at all.

How is the uniqueness theorem for power series related to the Taylor series?

The uniqueness theorem for power series is closely related to the Taylor series, as the Taylor series is a specific type of power series that represents a function within a certain interval. The uniqueness theorem guarantees that the Taylor series representation is unique within that interval.

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