Analytic functions
A function is analytic in one dimension around 0 if it can be written as a converging power series
Σ pₙ zⁿ. This definition can be extended to any dimension (even in infinite dimension) by
requiring that pₙ is a continuous n-multilinear map. In general, pₙ is not unique (in two
dimensions, taking p₂ (x, y) (x', y') = x y' or y x' gives the same map when applied to a
vector (x, y) (x, y)). A way to guarantee uniqueness is to take a symmetric pₙ, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach space is analytic, as well as the inverse on invertible operators.
Main definitions
Let p be a formal multilinear series from E to F, i.e., p n is a multilinear map on E^n
for n : ℕ.
p.radius: the largestr : ennrealsuch that∥p n∥ * r^ngrows subexponentially, defined as a liminf.p.le_radius_of_bound,p.bound_of_lt_radius,p.geometric_bound_of_lt_radius: relating the value of the radius with the growth of∥p n∥ * r^n.p.partial_sum n x: the sum∑_{i = 0}^{n-1} pᵢ xⁱ.p.sum x: the sum∑'_{i = 0}^{∞} pᵢ xⁱ.
Additionally, let f be a function from E to F.
has_fpower_series_on_ball f p x r: on the ball of centerxwith radiusr,f (x + y) = ∑'_n pₙ yⁿ.has_fpower_series_at f p x: on some ball of centerxwith positive radius, holdshas_fpower_series_on_ball f p x r.analytic_at 𝕜 f x: there exists a power seriespsuch that holdshas_fpower_series_at f p x.
We develop the basic properties of these notions, notably:
- If a function admits a power series, it is continuous (see
has_fpower_series_on_ball.continuous_onandhas_fpower_series_at.continuous_atandanalytic_at.continuous_at). - In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see
formal_multilinear_series.has_fpower_series_on_ball. - If a function admits a power series in a ball, then it is analytic at any point
yof this ball, and the power series there can be expressed in terms of the initial power seriespasp.change_origin y. Seehas_fpower_series_on_ball.change_origin. It follows in particular that the set of points at which a given function is analytic is open, seeis_open_analytic_at.
Implementation details
We only introduce the radius of convergence of a power series, as p.radius.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
The radius of a formal multilinear series
The radius of a formal multilinear series is the largest r such that the sum Σ pₙ yⁿ
converges for all ∥y∥ < r.
If ∥pₙ∥ rⁿ is bounded in n, then the radius of p is at least r.
For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ is bounded.
For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ tends to zero exponentially.
The radius of the sum of two formal series is at least the minimum of their two radii.
Given a formal multilinear series p and a vector x, then p.sum x is the sum Σ pₙ xⁿ. A
priori, it only behaves well when ∥x∥ < p.radius.
Given a formal multilinear series p and a vector x, then p.partial_sum n x is the sum
Σ pₖ xᵏ for k ∈ {0,..., n-1}.
Equations
- p.partial_sum n x = (finset.range n).sum (λ (k : ℕ), ⇑(p k) (λ (i : fin k), x))
The partial sums of a formal multilinear series are continuous.
Expanding a function as a power series
- r_le : r ≤ p.radius
- r_pos : 0 < r
- has_sum : ∀ {y : E}, y ∈ emetric.ball 0 r → has_sum (λ (n : ℕ), ⇑(p n) (λ (i : fin n), y)) (f (x + y))
Given a function f : E → F and a formal multilinear series p, we say that f has p as
a power series on the ball of radius r > 0 around x if f (x + y) = ∑' pₙ yⁿ for all ∥y∥ < r.
Given a function f : E → F and a formal multilinear series p, we say that f has p as
a power series around x if f (x + y) = ∑' pₙ yⁿ for all y in a neighborhood of 0.
Equations
- has_fpower_series_at f p x = ∃ (r : ennreal), has_fpower_series_on_ball f p x r
Given a function f : E → F, we say that f is analytic at x if it admits a convergent power
series expansion around x.
Equations
- analytic_at 𝕜 f x = ∃ (p : formal_multilinear_series 𝕜 E F), has_fpower_series_at f p x
If a function admits a power series expansion, then it is exponentially close to the partial sums of this power series on strict subdisks of the disk of convergence.
If a function admits a power series expansion at x, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., f (x + y)
is the uniform limit of p.partial_sum n y there.
If a function admits a power series expansion at x, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., f (x + y)
is the locally uniform limit of p.partial_sum n y there.
If a function admits a power series expansion at x, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., f y
is the uniform limit of p.partial_sum n (y - x) there.
If a function admits a power series expansion at x, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., f y
is the locally uniform limit of p.partial_sum n (y - x) there.
If a function admits a power series expansion on a disk, then it is continuous there.
In a complete space, the sum of a converging power series p admits p as a power series.
This is not totally obvious as we need to check the convergence of the series.
The sum of a converging power series is continuous in its disk of convergence.
Changing origin in a power series
If a function is analytic in a disk D(x, R), then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = sum_{n} p_n (y + z)^n = sum_{n, k} \choose n k p_n y^{n-k} z^k
= sum_{k} (sum_{n} \choose n k p_n y^{n-k}) z^k.
$$
The corresponding power series has thus a k-th coefficient equal to
\sum_{n} \choose n k p_n y^{n-k}. In the general case where pₙ is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets s of fin n of cardinal k, and attribute z to the indices in s and
y to the indices outside of s.
In this paragraph, we implement this. The new power series is called p.change_origin y. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
Changing the origin of a formal multilinear series p, so that
p.sum (x+y) = (p.change_origin x).sum y when this makes sense.
Here, we don't use the bracket notation ⟨n, s, hs⟩ in place of the argument i in the lambda,
as this leads to a bad definition with auxiliary _match statements,
but we will try to use pattern matching in lambdas as much as possible in the proofs below
to increase readability.
Auxiliary lemma controlling the summability of the sequence appearing in the definition of
p.change_origin, first version.
Auxiliary lemma controlling the summability of the sequence appearing in the definition of
p.change_origin, second version.
An auxiliary definition for change_origin_radius.
Auxiliary lemma controlling the summability of the sequence appearing in the definition of
p.change_origin, third version.
The radius of convergence of p.change_origin x is at least p.radius - ∥x∥. In other words,
p.change_origin x is well defined on the largest ball contained in the original ball of
convergence.
The k-th coefficient of p.change_origin is the sum of a summable series.
Summing the series p.change_origin x at a point y gives back p (x + y)
If a function admits a power series expansion p on a ball B (x, r), then it also admits a
power series on any subball of this ball (even with a different center), given by p.change_origin.