|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of identity theorem of power series
|
(Proof)
|
|
|
We can prove the identity theorem for power series using divided differences. From amongst the points at which the two series are equal, pick a sequence $\{w_k\}_{k=0}^\infty$ which satisfies the following three conditions:
- $\lim_{k \to \infty} w_k = z_0$
- $w_m = w_n$ if and only if $m = n$ .
- $w_k \neq z_0$ for all $k$ .
Let $f$ be the function determined by one power series and let $g$ be the function determined by the other power series:
Because formation of divided differences involves finite sums and dividing by differences of $w_k$ 's (which all differ from zero by condition 2 above, so it is legitimate to divide by them), we may carry out the formation of finite diffferences on a term-by-term basis. Using the result about divided differences of powers, we have
where$$ D_{mnk} = \sum_{j_0 + \ldots j_m = n - m} (w_k - z_0)^{j_0} \cdots (w_{k+m} - z_0)^{j_m}.$$
Note that $\lim_{k \to infty} D_{mnk} = 0$ when $m > n$ , but $D_{mmk} = 1$ . Since power series converge uniformly, we may intechange limit and summation to conclude
Since, by design, $f(w_k) = g(w_k)$ , we have$$ \Delta^m f [w_k, \ldots, w_{k+m}] = \Delta^m g [w_k, \ldots, w_{k+m}],$$ hence $a_m = b_m$ for all $m$ .
|
"proof of identity theorem of power series" is owned by rspuzio.
|
|
(view preamble | get metadata)
Cross-references: design, limit, converge, divided differences of powers, basis, divide, differences, sums, finite, function, sequence, series, points, divided differences, power series, identity theorem
This is version 8 of proof of identity theorem of power series, born on 2007-03-08, modified 2007-03-08.
Object id is 9047, canonical name is ProofOfIdentityTheoremOfPowerSeries2.
Accessed 1630 times total.
Classification:
| AMS MSC: | 30B10 (Functions of a complex variable :: Series expansions :: Power series ) | | | 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle \Delta^m f [w_k, \ldots, w_{k+m}]$](http://images.planetmath.org:8080/cache/objects/9047/js/img5.png "$\displaystyle \Delta^m f [w_k, \ldots, w_{k+m}]$"){kind=small}
![$\displaystyle \Delta^m f [w_k, \ldots, w_{k+m}]$](http://images.planetmath.org:8080/cache/objects/9047/js/img7.png "$\displaystyle \Delta^m f [w_k, \ldots, w_{k+m}]$"){kind=small}
![$\displaystyle \lim_{k \to \infty} \Delta^m f [w_k, \ldots, w_{k+m}]$](http://images.planetmath.org:8080/cache/objects/9047/js/img9.png "$\displaystyle \lim_{k \to \infty} \Delta^m f [w_k, \ldots, w_{k+m}]$"){kind=small}
![$\displaystyle \lim_{k \to \infty} \Delta^m g [w_k, \ldots, w_{k+m}]$](http://images.planetmath.org:8080/cache/objects/9047/js/img11.png "$\displaystyle \lim_{k \to \infty} \Delta^m g [w_k, \ldots, w_{k+m}]$"){kind=small}
