analytic_struct_operation

[analytic_struct_product] Product space

Asymmetry: ==> No

We have to use and partial derivatives

something like

[mulitplication_analytic]

with

The radius of convergence is at least

(Related to Cauchy product. Try to find a better proof method)

Restore factor in differentiation,

==> [Leibniz_law_1d]

, or

Collect the tensor

let to get the polynomial

==>

==> [Leibniz_law]

[mulitplication_inverse_analytic]

let , ,

use and

The convergence radius of is controlled by the convergence radius of , which is .

or calculate directly

let , , use multiplication

==> , use induction

[differential_of_multiplication_inverse] use Leibniz law

, or

in particular,

radius of convergence (of )

choose so that

try inductive proof , correspond to radius of convergence

first calculate

then calculate

this give the start of induction

try inductive proof

[compose_op_analytic]

let ,

with

where all possible sources of the compounded

with

thus can only come from for

(cf. multi_combination)

==> (Faà di Bruno formula)

. Written as a differential [chain_rule_1d]

, or

where

, written as a differential is [chain_rule]

Generally written as the differential form

Extract

Place in

Get (this is not )

Faà di Bruno formula

[inverse_analytic]

let , ,

let

First-order differential calculation. , use composite

Higher-order differential calculation. use induction for

only comes from

and ==>

==> (omitting )

==>

Because it may not converge, cannot be directly used as a function

But it can be extended to

such that

The radius of convergence of the inverse function is non-zero (p.77 of ref-4)

==>

use (indeed )

Construct a power series control with a non-zero radius of convergence for (almost) such that

if by induction, for , , ,

where with

Its inverse is with . to prove. radius of convergence is non-zero to prove

use case of

to get , use

==>

to get , , use

to get , use

now prove the inverse power series of the power series has a non-zero radius of convergence

let ,

In order to find the inverse mapping of , solve the equation

==> Quadratic equation of , there are two roots

use , select the correct root

use radius of convergence ==> non-zero radius of convergence

use ==> non-zero radius of convergence

Although the exact radius of convergence cannot be given here, the method of proving the inverse function by the compression fixed point principle cannot give the exact maximal local reversible region for the pure differential method.

Question Both methods seem very ad hoc? Is there a method more inspired by the intuition of the inverse function itself? For example, can it be related to the “analyticity” of the mapping ?

[differential_of_inverse]

[implicit_function]

use analytic_struct_product

and

==> ,

The calculation of differentials and differential functions does not require series in advance

Analytic functions are not closed under non-finite summation. Using trigonometric or exponential functions, series can provide things that are discontinuous, continuous but non-differentiable, or differentiable but non-analytic

function with zero radius of convergence at finite points

connected to
Function that is everywhere but has a radius of convergence of everywhere

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

Continuous but nowhere differentiable

wiki: Weierstrass_function

where , is positive odd integer, and

-th order differentiable but not -th order differentiable: use the integrals of each order of the Weierstrass function
-th order differentiable but -th order not continuously differentiable (although -th order differentiable implies -th order continuously differentiable): use , -th order differentiable but not -th order continuously differentiable, use its integrals of each order
Continuous homeomorphism but not differentiable homeomorphism or analytic homeomorphism.
Diffeomorphism but not analytic diffeomorphism. Take the part of the smooth but everywhere non-analytic function where to get a local diffeomorphism. Local to global by using to get an analytic diffeomorphism from

note-math