解析 struct 的操作

[analytic-struct-product] 积空间

不对称性: ==> 没有

只好使用和偏微分

something like

[mulitplication-analytic]

with

收敛半径至少是

(联系于 Cauchy product. 尝试寻找更好的证明方法)

恢复微分中的 ,

==> [Leibniz-law-1d]

, or

收集重张量

let 得到多项式

==>

==> [Leibniz-law]

[mulitplication-inverse-analytic]

let , ,

use and

的收敛半径受到的收敛半径的控制

or 直接计算

let , , use 乘法

==> , use induction

[differential-of-multiplication-inverse] use Leibniz law

, or

in particular,

收敛半径

try 归纳证明

为完成归纳, use with

[compose-op-analytic]

let ,

with

where 复合后的的所有可能来源

with

从而只能来自 for

(cf. multi-combination)

==>

. 写为微分 [chain-rule-1d]

, or

where

, 写为微分就是 [chain-rule]

一般的写为微分的形式

将提取

置于

得到 (this is not )

[inverse-analytic]

let , ,

let

一阶微分计算. , use 复合

高阶微分计算. use 归纳 for

只来自

and ==>

==> (省略 )

==>

由于可能不收敛, 无法直接作为函数

但是可以扩充到

使得

逆函数的收敛半径非零 (p.77 of ref-4)

==>

use (indeed )

构造 (几乎) 的非零收敛半径的幂级数控制

if by induction, for , , ,

where with

其逆是 with . to prove. 收敛半径非零 to prove

use case of

to get , use

==>

to get , , use

to get , use

now prove 幂级数的逆幂级数收敛半径非零

let ,

为了求的逆映射 , 解方程

==> 的二次方程, 有两个根

use , 选取正确的根

use 收敛半径 ==> 非零收敛半径

use ==> 非零收敛半径

虽然这里无法给出确切的收敛半径, 但是对于纯微分方法, 压缩不动点原理证明逆函数的方法也不能给出确切的极大局部可逆区域

Question 其实两种方法看起来都很 ad hoc? 有无更受到逆函数本身的直观所启发的方法? 例如是否能联系到映射的 "解析性"?

[differential-of-inverse]

[implicit-function]

use analytic-struct-product

and

==> ,

微分和微分函数的计算不需要预先有级数

有限点处收敛半径为零的函数

接上
处处但处处收敛半径的函数

wiki: Non-analytic_smooth_function

Since the series converges for forall , this function is easily seen to be of class , by a standard inductive application of the Weierstrass M-test to demonstrate uniform convergence of each series of derivatives.

We now show that is not analytic at any dyadic rational multiple of , that is, at any with and .

Since the sum of the first q terms is analytic, we need only consider , the sum of the terms with .

For forall orders of derivation with , and we have

where we used the fact that for forall , and we bounded the first sum from below by the term with .

As a consequence, at any such ,

Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that , and hence , is nowhere analytic in

连续但处处不可微

wiki: Weierstrass_function

where , is positive odd integer, and

阶可微但不阶可微: 使用 Weierstrass 函数的各阶积分
阶可微但阶不连续可微 (虽然阶可微蕴含阶连续可微): 使用 , 阶可微但不阶连续可微, 使用其各阶积分
连续同胚但不微分同胚不解析同胚.
微分同胚但不解析同胚. 光滑但处处不解析函数中取的部分得到局部微分同胚. 局部到全局 by 用得到的解析同胚

note-math