analytic_Euclidean

Starting from the one-dimensional case

Geometric series

in ,

[convergence_radius_1d] Radius of convergence

(cf. limsup)

==>

[absolute_convergence_analytic_1d]

Prop ==> converges absolutely

Proof

and

use Geometric series test and

Prop ==> diverges absolutely

Proof ==> For infinitely many ,

Prop converges absolutely ==>

[uniformaly_absolutely_convergence_analytic]

use . use Geometric series domination

In the closed ball of radius , converges uniformly and absolutely

The polynomial function is continuous

Within the radius of convergence, the function defined by the power series

[analytic_imply_continuous]

==> continuous

Example

The radius of convergence for is
The radius of convergence for is

Convergence issues on the boundary

The radius of convergence for is , at it is the harmonic series , which diverges absolutely
The radius of convergence for is , at it converges absolutely to
Absolute convergence vs. convergence: converges for , but not absolutely

Generalize the change_base_point_polynomial of polynomials to series

[change_base_point_analytic]

==> Power series after shifting the base point to

It also has a non-zero radius of convergence at . By the triangle inequality,

It converges absolutely when , i.e., , thus

Now consider the higher-dimensional case. Power series

Note the symmetry, e.g., for , for

Generalize the polynomial function polynomial_function to power series

Unlike the one-dimensional case, in higher dimensions, generally does not hold. Even is not yet defined

[linear_map_induced_norm]

let

is defined as the uniform control coefficient for all directions . The compactness of will make this definition meaningful

so that (for all direction)

and

Compared to the case, the computability of the definition for is lower

[convergence_radius] Convergence radius

[absolute_convergence_analytic]

same as

==> converges absolutely
There exists a direction , forall , diverges absolutely

Proof (of divergence)

Use linear_map_induced_norm , there exists such that

Using the definition, for infinitely many ,

use passing to compact and subsequence converges to

==> infinitely many terms in

scale to

==>

let

==> infinitely many terms in

Prop converges absolutely ==>

similar to one-dimensional case, also have

uniformaly_absolutely_convergence_analytic
analytic_imply_continuous
change_base_point_analytic

for , the -th order difference gives

substitute

power series converges uniformly absolutely within the radius of convergence, thus limits can be interchanged

can recover the -th order monomial

[differential]

-th order differential

Example

the definitions of difference and differential can be applied to any function, not necessarily defined by power series

[polynomial_expansion] Polynomial expansion

alias power series, Taylor expansion, Taylor series

[polynomial_approximation] Polynomial approximation

alias Taylor expansion, Taylor approximation, Taylor polynomial [Taylor_expansion] [Taylor_approximation] [Taylor_polynomial]

[derivative] Derivative alias derivative, directional derivative

Successive differences and derivatives

Successive difference Independent of order + limit exchange ==> Commutativity of directional derivatives

[successive_derivative] Successive derivative

==> Directional derivative representation of power series

The concept of successive derivative uses the subtraction of tangent vectors at different points, implicitly employing the concept of connection

[partial_derivative] Partial derivative

Using coordinates. let be the basis of . so coordinate component

and so on

let . use successive_derivative, partial_derivative

==> Partial derivative representation of power series (also cf. multi_combination)

when domain = ,

define and dual basis with

==> Partial derivative representation of differential as coefficient–basis expansion of symmetric tensor

when domain =

Example

let

, or

if using range space coordinates then first-order differential is represented as Jacobi matrix [Jacobi_matrix]

[differential_function] Differential function

Treating the range as a linear space, using the power norm, allows for power series expansion

[successive_differential]

isomorphism

with

same norm

same convergence radius (use )

Proof (draft) Commutativity of derivatives and . norm estimation

Abbreviation despite notational conflict

==> Power series of differential function

[anti_derivative]

use

==> . Zero-order term is indeterminate
…

[mean_value_theorem_analytic_1d] Mean Value Theorem for Differentiation

Intermediate value ver. for function
compact uniform linear growth control ver.

Proof

use reduce to

Both cases

thus has extremum and . Then

[fundamental_theorem_of_calculus] Fundamental Theorem of Calculus

Technique used in proof: Mean Value Theorem compact uniform linear growth control ver. + compact partition uniform approximation

[mean_value_theorem_analytic] Higher dimensions generally lack intermediate value ver. Mean Value Theorem for . Use embedded line reduce to case

First order

by the Fundamental Theorem of Calculus and chain_rule_1d and

remainder estimation, uniform linear control

higher order

by integration by parts

summed from up to

remainder estimation, uniform order power control

let power series

[convergence_domain] convergence domain at a point :=

computing the coefficients after changing the base point of a power series uses the interchange of summation

for polynomials, the sum is finite, the order of summation can be interchanged, thus changing the base point is well-defined change_base_point_polynomial

however, for infinite sums (limits), if not absolutely convergent, they are not always compatible with changes in summation order series_rearrangement

changing the base point of a power series may alter the convergence domain

Example

with

convergence domain is

changing the base point leads to a change in the convergence domain

, ,

convergence domain , an open ball of radius
,

convergence domain , an open ball of radius

repeatedly changing the base point can “alter” the value it converges to

Example

let with

let successively switch base points , finally returning to

if each displacement is within the convergence region of the base point

then the final power series is , where is the number of times the path formed by (counterclockwise) winds around

. Winding around times yields

Winding around times yields , by

[analytic_continuation]

Well-defined continuation region: unaffected by switching base points
Maximal continuation region: cannot be continued well-definedly any further

Example

convergence radius

Cannot be continued well-definedly to . by winding around times yields

The maximal well-defined continuation region should be

convergence radius

Can be well-definedly extended to , coinciding with defined by division in

Note , or . Indicates that derivative or antiderivative affects

and are already maximal extensions

The maximal extension of is

The power series coefficients of contain complex numbers, unlike which only contains real numbers

[analytic_function] Analytic function := For every point in the domain of , can be defined near by a power series at : . Here

[analytic_isomorphism] Analytic isomorphism :=

is bijective
are analytic functions

This implies , because

Example

is an analytic isomorphism.

==> , monotonically increasing ==> is an analytic diffeomorphism

, in has solutions ==> ==> is not an analytic diffeomorphism

with is an analytic diffeomorphism

with is a local analytic diffeomorphism, but not an analytic diffeomorphism. Not injective:

[power_series_space]

Power series space

Attempt to define a distance on the power series space. Expect to be close within some radius , in other words, is close to within radius

(note: is linear_map_induced_norm

let

Note we performed a radius truncation , at this point on the closed disk of radius , the power series converges absolutely and uniformly

closed disk is compact, which brings many good properties. Consider , it is unbounded near . Then for , no matter how close is to , is still unbounded near . But if we consider the closed disk of radius centered at the origin , there is bounded

Example The truncated polynomials (Taylor polynomials) of the power series itself also approximate . Because

Another possibly topologically equivalent formulation is to use . The equivalence is because

Take , then

There is a possibly too weak topology. .

Let . Although and the radii of convergence for are both . The value of at is , the value of at is . In this case is also

There is a possibly too strong topology

or Based on the given , it should be possible to construct satisfying such conditions, at least the case is simple

Define the distance between power series

As a uniform control for

It is not a norm.

Why is this topology said to be too strong? Consider the case , consider , then

Should it be ?

Under this definition of distance, no matter how close is to

This means that this topology is too strong in the sense that . The reason might be that the inequality is too crude. By raising both sides to the power of and comparing, one can see

Prop

Proof For , take .

Now consider the topology of the space of analytic functions. We need to use techniques similar to the compact-open topology used for spaces of continuous functions

The radius of convergence of an analytic function at each point should be a continuous function

Let compact be contained in the domain of the analytic function . Then has a non-zero infimum on the compact set . That is, . Therefore, we can define the norm of on as

If there exists for a compact such that , then by the definition of analyticity, is analytic on

For an open set , is analytic on every compact <==> is analytic on

For compact , define the space (it is a Banach space)

The topological basis or net basis of , defined as

is expressed as

where can be replaced by any net structure beyond

Prop

Proof Fix . Take to get . Take , take such that

==>

Prop The Taylor polynomial of expanded at converges to on

Proof

let

Take ? Obtain and then

Prop For real analytic functions, the zero-order cannot control

Example . . . Since , it is impossible that

If a real analytic function is extended to a complex analytic function (by extending from to ), then by the Cauchy integral formula it can be proven that the topology is equivalent to , where

Note that, the zero-order control for in non-real space, if one wants to express it through the real functions and , requires control of the higher-order derivatives of the real functions

Take the one-dimensional case as an example. let . let

Thus , i.e.

let , let . let

Thus

Thus , i.e.

Example

(if )

in analytic spaces and their nets

[inverse_op_continous_in_analytic_space] ==>
[compose_op_continous_in_analytic_space] and ==>

Or rather, operators are all continuous functions of analytic spaces

same for linear , multiplication , inversion ?

We need to estimate . We prove is a Banach algebra

Therefore

Assume is nonzero on , then is also analytic. Considering that and may have different convergence properties, if necessary, shrink . Then by the triangle inequality and multiplication inequality of the norm

It’s enough to choose ?

Composition , compositional inverse . Omitted for now

Connected components of the topology of analytic function spaces

[homotopy_analytic] Analytic homotopy

[power_series_analytic_equivalent] Analytically equivalent power series := Two power series come from the power series expansion of the same analytic function at different points. Is this equivalent to all possible analytic continuations? (Riemann Surface?)

[power_series_analytic_homotopy_equivalent] Analytically homotopy equivalent power series := Two power series come from the power series expansion of the same analytic function homotopy class at different points

note-math