# mathlibdocumentation

analysis.analytic.basic

# 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 largest r : ℝ≥0∞ such that ∥p n∥ * r^n grows subexponentially.
• p.le_radius_of_bound, p.le_radius_of_bound_nnreal, p.le_radius_of_is_O: if ∥p n∥ * r ^ n is bounded above, then r ≤ p.radius;
• p.is_o_of_lt_radius, p.norm_mul_pow_le_mul_pow_of_lt_radius, p.is_o_one_of_lt_radius, p.norm_mul_pow_le_of_lt_radius, p.nnnorm_mul_pow_le_of_lt_radius: if r < p.radius, then ∥p n∥ * r ^ n tends to zero exponentially;
• p.lt_radius_of_is_O: if r ≠ 0 and ∥p n∥ * r ^ n = O(a ^ n) for some -1 < a < 1, then r < p.radius;
• 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 center x with radius r, f (x + y) = ∑'_n pₙ yⁿ.
• has_fpower_series_at f p x: on some ball of center x with positive radius, holds has_fpower_series_on_ball f p x r.
• analytic_at 𝕜 f x: there exists a power series p such that holds has_fpower_series_at f p x.
• analytic_on 𝕜 f s: the function f is analytic at every point of s.

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_on and has_fpower_series_at.continuous_at and analytic_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 y of this ball, and the power series there can be expressed in terms of the initial power series p as p.change_origin y. See has_fpower_series_on_ball.change_origin. It follows in particular that the set of points at which a given function is analytic is open, see is_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.

@[protected]
noncomputable def formal_multilinear_series.sum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ring 𝕜] [ E] [ F] (p : F) (x : E) :
F

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.

Equations
def formal_multilinear_series.partial_sum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ring 𝕜] [ E] [ F] (p : F) (n : ) (x : E) :
F

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
theorem formal_multilinear_series.partial_sum_continuous {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ring 𝕜] [ E] [ F] (p : F) (n : ) :

The partial sums of a formal multilinear series are continuous.

### The radius of a formal multilinear series #

noncomputable def formal_multilinear_series.radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) :

The radius of a formal multilinear series is the largest r such that the sum Σ ∥pₙ∥ ∥y∥ⁿ converges for all ∥y∥ < r. This implies that Σ pₙ yⁿ converges for all ∥y∥ < r, but these definitions are not equivalent in general.

Equations
theorem formal_multilinear_series.le_radius_of_bound {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (C : ) {r : nnreal} (h : ∀ (n : ), p n * r ^ n C) :

If ∥pₙ∥ rⁿ is bounded in n, then the radius of p is at least r.

theorem formal_multilinear_series.le_radius_of_bound_nnreal {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (C : nnreal) {r : nnreal} (h : ∀ (n : ), p n∥₊ * r ^ n C) :

If ∥pₙ∥ rⁿ is bounded in n, then the radius of p is at least r.

theorem formal_multilinear_series.le_radius_of_is_O {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : (λ (n : ), p n * r ^ n) =O[filter.at_top] λ (n : ), 1) :

If ∥pₙ∥ rⁿ = O(1), as n → ∞, then the radius of p is at least r.

theorem formal_multilinear_series.le_radius_of_eventually_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (C : ) (h : ∀ᶠ (n : ) in filter.at_top, p n * r ^ n C) :
theorem formal_multilinear_series.le_radius_of_summable_nnnorm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : summable (λ (n : ), p n∥₊ * r ^ n)) :
theorem formal_multilinear_series.le_radius_of_summable {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : summable (λ (n : ), p n * r ^ n)) :
theorem formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (h : ∀ (r : nnreal), (λ (n : ), p n * r ^ n) =O[filter.at_top] λ (n : ), 1) :
theorem formal_multilinear_series.radius_eq_top_of_eventually_eq_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (h : ∀ᶠ (n : ) in filter.at_top, p n = 0) :
theorem formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (n : ) (hn : ∀ (m : ), p (m + n) = 0) :
theorem formal_multilinear_series.is_o_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
∃ (a : ) (H : a 1), (λ (n : ), p n * r ^ n) =o[filter.at_top]

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ tends to zero exponentially: for some 0 < a < 1, ∥p n∥ rⁿ = o(aⁿ).

theorem formal_multilinear_series.is_o_one_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
(λ (n : ), p n * r ^ n) =o[filter.at_top] λ (_x : ), 1

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ = o(1).

theorem formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
∃ (a : ) (H : a 1) (C : ) (H : C > 0), ∀ (n : ), p n * r ^ n C * a ^ n

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ tends to zero exponentially: for some 0 < a < 1 and C > 0, ∥p n∥ * r ^ n ≤ C * a ^ n.

theorem formal_multilinear_series.lt_radius_of_is_O {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h₀ : r 0) {a : } (ha : a set.Ioo (-1) 1) (hp : (λ (n : ), p n * r ^ n) =O[filter.at_top] ) :

If r ≠ 0 and ∥pₙ∥ rⁿ = O(aⁿ) for some -1 < a < 1, then r < p.radius.

theorem formal_multilinear_series.norm_mul_pow_le_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
∃ (C : ) (H : C > 0), ∀ (n : ), p n * r ^ n C

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ is bounded.

theorem formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h0 : 0 < r) (h : r < p.radius) :
∃ (C : ) (H : C > 0), ∀ (n : ), p n C / r ^ n

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ is bounded.

theorem formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
∃ (C : nnreal) (H : C > 0), ∀ (n : ), p n∥₊ * r ^ n C

For r strictly smaller than the radius of p, then ∥pₙ∥ rⁿ is bounded.

theorem formal_multilinear_series.le_radius_of_tendsto {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {r : nnreal} (p : F) {l : } (h : filter.tendsto (λ (n : ), p n * r ^ n) filter.at_top (nhds l)) :
theorem formal_multilinear_series.le_radius_of_summable_norm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {r : nnreal} (p : F) (hs : summable (λ (n : ), p n * r ^ n)) :
theorem formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} (h : p.radius < x∥₊) :
¬summable (λ (n : ), p n * x ^ n)
theorem formal_multilinear_series.summable_norm_mul_pow {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
summable (λ (n : ), p n * r ^ n)
theorem formal_multilinear_series.summable_norm_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} (hx : x ) :
summable (λ (n : ), (p n) (λ (_x : fin n), x))
theorem formal_multilinear_series.summable_nnnorm_mul_pow {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (h : r < p.radius) :
summable (λ (n : ), p n∥₊ * r ^ n)
@[protected]
theorem formal_multilinear_series.summable {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} (hx : x ) :
summable (λ (n : ), (p n) (λ (_x : fin n), x))
theorem formal_multilinear_series.radius_eq_top_of_summable_norm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (hs : ∀ (r : nnreal), summable (λ (n : ), p n * r ^ n)) :
theorem formal_multilinear_series.radius_eq_top_iff_summable_norm {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) :
p.radius = ∀ (r : nnreal), summable (λ (n : ), p n * r ^ n)
theorem formal_multilinear_series.le_mul_pow_of_radius_pos {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (h : 0 < p.radius) :
∃ (C r : ) (hC : 0 < C) (hr : 0 < r), ∀ (n : ), p n C * r ^ n

If the radius of p is positive, then ∥pₙ∥ grows at most geometrically.

theorem formal_multilinear_series.min_radius_le_radius_add {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p q : F) :

The radius of the sum of two formal series is at least the minimum of their two radii.

@[simp]
theorem formal_multilinear_series.radius_neg {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) :
@[protected]
theorem formal_multilinear_series.has_sum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} (hx : x ) :
has_sum (λ (n : ), (p n) (λ (_x : fin n), x)) (p.sum x)
theorem formal_multilinear_series.radius_le_radius_continuous_linear_map_comp {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] (p : F) (f : F →L[𝕜] G) :

### Expanding a function as a power series #

structure has_fpower_series_on_ball {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E → F) (p : F) (x : E) (r : ennreal) :
Prop

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.

def has_fpower_series_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E → F) (p : F) (x : E) :
Prop

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
def analytic_at (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E → F) (x : E) :
Prop

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
• f x = ∃ (p : , x
def analytic_on (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E → F) (s : set E) :
Prop

Given a function f : E → F, we say that f is analytic on a set s if it is analytic around every point of s.

Equations
• f s = ∀ (x : E), x s f x
theorem has_fpower_series_on_ball.has_fpower_series_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
x
theorem has_fpower_series_at.analytic_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) :
f x
theorem has_fpower_series_on_ball.analytic_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
f x
theorem has_fpower_series_on_ball.congr {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {p : F} {x : E} {r : ennreal} (hf : r) (hg : g r)) :
r
theorem has_fpower_series_on_ball.comp_sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) (y : E) :
has_fpower_series_on_ball (λ (z : E), f (z - y)) p (x + y) r

If a function f has a power series p around x, then the function z ↦ f (z - y) has the same power series around x + y.

theorem has_fpower_series_on_ball.has_sum_sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) {y : E} (hy : y r) :
has_sum (λ (n : ), (p n) (λ (i : fin n), y - x)) (f y)
theorem has_fpower_series_on_ball.radius_pos {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
theorem has_fpower_series_at.radius_pos {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) :
theorem has_fpower_series_on_ball.mono {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r r' : ennreal} (hf : r) (r'_pos : 0 < r') (hr : r' r) :
r'
@[protected]
theorem has_fpower_series_at.eventually {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) :
∀ᶠ (r : ennreal) in (set.Ioi 0), r
theorem has_fpower_series_on_ball.add {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {pf pg : F} {x : E} {r : ennreal} (hf : r) (hg : r) :
(pf + pg) x r
theorem has_fpower_series_at.add {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {pf pg : F} {x : E} (hf : x) (hg : x) :
has_fpower_series_at (f + g) (pf + pg) x
theorem analytic_at.add {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {x : E} (hf : f x) (hg : g x) :
(f + g) x
theorem has_fpower_series_on_ball.neg {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {pf : F} {x : E} {r : ennreal} (hf : r) :
(-pf) x r
theorem has_fpower_series_at.neg {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {pf : F} {x : E} (hf : x) :
(-pf) x
theorem analytic_at.neg {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {x : E} (hf : f x) :
(-f) x
theorem has_fpower_series_on_ball.sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {pf pg : F} {x : E} {r : ennreal} (hf : r) (hg : r) :
(pf - pg) x r
theorem has_fpower_series_at.sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {pf pg : F} {x : E} (hf : x) (hg : x) :
has_fpower_series_at (f - g) (pf - pg) x
theorem analytic_at.sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f g : E → F} {x : E} (hf : f x) (hg : g x) :
(f - g) x
theorem has_fpower_series_on_ball.coeff_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {pf : F} {x : E} {r : ennreal} (hf : r) (v : fin 0 → E) :
(pf 0) v = f x
theorem has_fpower_series_at.coeff_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {pf : F} {x : E} (hf : x) (v : fin 0 → E) :
(pf 0) v = f x
theorem continuous_linear_map.comp_has_fpower_series_on_ball {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] {f : E → F} {p : F} {x : E} {r : ennreal} (g : F →L[𝕜] G) (h : r) :
x r

If a function f has a power series p on a ball and g is linear, then g ∘ f has the power series g ∘ p on the same ball.

theorem continuous_linear_map.comp_analytic_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [ E] [ F] [ G] {f : E → F} {s : set E} (g : F →L[𝕜] G) (h : f s) :
(g f) s

If a function f is analytic on a set s and g is linear, then g ∘ f is analytic on s.

theorem has_fpower_series_on_ball.uniform_geometric_approx' {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} {r' : nnreal} (hf : r) (h : r' < r) :
∃ (a : ) (H : a 1) (C : ) (H : C > 0), ∀ (y : E), y r'∀ (n : ), f (x + y) - p.partial_sum n y C * (a * (y / r')) ^ n

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.

This version provides an upper estimate that decreases both in ∥y∥ and n. See also has_fpower_series_on_ball.uniform_geometric_approx for a weaker version.

theorem has_fpower_series_on_ball.uniform_geometric_approx {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} {r' : nnreal} (hf : r) (h : r' < r) :
∃ (a : ) (H : a 1) (C : ) (H : C > 0), ∀ (y : E), y r'∀ (n : ), f (x + y) - p.partial_sum n y C * a ^ n

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.

theorem has_fpower_series_at.is_O_sub_partial_sum_pow {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) (n : ) :
(λ (y : E), f (x + y) - p.partial_sum n y) =O[nhds 0] λ (y : E), y ^ n

Taylor formula for an analytic function, is_O version.

theorem has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r r' : ennreal} (hf : r) (hr : r' < r) :
(λ (y : E × E), f y.fst - f y.snd - (p 1) (λ (_x : fin 1), y.fst - y.snd)) =O[filter.principal (emetric.ball (x, x) r')] λ (y : E × E), y - (x, x) * y.fst - y.snd

If f has formal power series ∑ n, pₙ on a ball of radius r, then for y, z in any smaller ball, the norm of the difference f y - f z - p 1 (λ _, y - z) is bounded above by C * (max ∥y - x∥ ∥z - x∥) * ∥y - z∥. This lemma formulates this property using is_O and filter.principal on E × E.

theorem has_fpower_series_on_ball.image_sub_sub_deriv_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r r' : ennreal} (hf : r) (hr : r' < r) :
∃ (C : ), ∀ (y : E), y r'∀ (z : E), z r'f y - f z - (p 1) (λ (_x : fin 1), y - z) C * z - x * y - z

If f has formal power series ∑ n, pₙ on a ball of radius r, then for y, z in any smaller ball, the norm of the difference f y - f z - p 1 (λ _, y - z) is bounded above by C * (max ∥y - x∥ ∥z - x∥) * ∥y - z∥.

theorem has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) :
(λ (y : E × E), f y.fst - f y.snd - (p 1) (λ (_x : fin 1), y.fst - y.snd)) =O[nhds (x, x)] λ (y : E × E), y - (x, x) * y.fst - y.snd

If f has formal power series ∑ n, pₙ at x, then f y - f z - p 1 (λ _, y - z) = O(∥(y, z) - (x, x)∥ * ∥y - z∥) as (y, z) → (x, x). In particular, f is strictly differentiable at x.

theorem has_fpower_series_on_ball.tendsto_uniformly_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} {r' : nnreal} (hf : r) (h : r' < r) :
tendsto_uniformly_on (λ (n : ) (y : E), p.partial_sum n y) (λ (y : E), f (x + y)) filter.at_top r')

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.

theorem has_fpower_series_on_ball.tendsto_locally_uniformly_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
tendsto_locally_uniformly_on (λ (n : ) (y : E), p.partial_sum n y) (λ (y : E), f (x + y)) filter.at_top r)

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.

theorem has_fpower_series_on_ball.tendsto_uniformly_on' {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} {r' : nnreal} (hf : r) (h : r' < r) :
tendsto_uniformly_on (λ (n : ) (y : E), p.partial_sum n (y - x)) f filter.at_top r')

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.

theorem has_fpower_series_on_ball.tendsto_locally_uniformly_on' {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
tendsto_locally_uniformly_on (λ (n : ) (y : E), p.partial_sum n (y - x)) f filter.at_top r)

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.

@[protected]
theorem has_fpower_series_on_ball.continuous_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
r)

If a function admits a power series expansion on a disk, then it is continuous there.

@[protected]
theorem has_fpower_series_at.continuous_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} (hf : x) :
@[protected]
theorem analytic_at.continuous_at {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {x : E} (hf : f x) :
@[protected]
theorem analytic_on.continuous_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {s : set E} (hf : f s) :
@[protected]
theorem formal_multilinear_series.has_fpower_series_on_ball {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (h : 0 < p.radius) :

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.

theorem has_fpower_series_on_ball.sum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (h : r) {y : E} (hy : y r) :
f (x + y) = p.sum y
@[protected]
theorem formal_multilinear_series.continuous_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {p : F}  :

The sum of a converging power series is continuous in its disk of convergence.

### Uniqueness of power series #

If a function f : E → F has two representations as power series at a point x : E, corresponding to formal multilinear series p₁ and p₂, then these representations agree term-by-term. That is, for any n : ℕ and y : E, p₁ n (λ i, y) = p₂ n (λ i, y). In the one-dimensional case, when f : 𝕜 → E, the continuous multilinear maps p₁ n and p₂ n are given by formal_multilinear_series.mk_pi_field, and hence are determined completely by the value of p₁ n (λ i, 1), so p₁ = p₂. Consequently, the radius of convergence for one series can be transferred to the other.

theorem asymptotics.is_O.continuous_multilinear_map_apply_eq_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {n : } {p : (λ (i : fin n), E) F} (h : (λ (y : E), p (λ (i : fin n), y)) =O[nhds 0] λ (y : E), y ^ (n + 1)) (y : E) :
p (λ (i : fin n), y) = 0
theorem has_fpower_series_at.apply_eq_zero {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {p : F} {x : E} (h : x) (n : ) (y : E) :
(p n) (λ (i : fin n), y) = 0

If a formal multilinear series p represents the zero function at x : E, then the terms p n (λ i, y) appearing the in sum are zero for any n : ℕ, y : E.

theorem has_fpower_series_at.eq_zero {𝕜 : Type u_1} {E : Type u_2} [ E] {p : E} {x : 𝕜} (h : x) :
p = 0

A one-dimensional formal multilinear series representing the zero function is zero.

theorem has_fpower_series_at.eq_formal_multilinear_series {𝕜 : Type u_1} {E : Type u_2} [ E] {p₁ p₂ : E} {f : 𝕜 → E} {x : 𝕜} (h₁ : x) (h₂ : x) :
p₁ = p₂

One-dimensional formal multilinear series representing the same function are equal.

theorem has_fpower_series_on_ball.exchange_radius {𝕜 : Type u_1} {E : Type u_2} [ E] {p₁ p₂ : E} {f : 𝕜 → E} {r₁ r₂ : ennreal} {x : 𝕜} (h₁ : r₁) (h₂ : r₂) :
r₂

If a function f : 𝕜 → E has two power series representations at x, then the given radii in which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear series in one representation has a particularly nice form, but the other has a larger radius.

theorem has_fpower_series_on_ball.r_eq_top_of_exists {𝕜 : Type u_1} {E : Type u_2} [ E] {f : 𝕜 → E} {r : ennreal} {x : 𝕜} {p : E} (h : r) (h' : ∀ (r' : nnreal), 0 < r'(∃ (p' : , r')) :

If a function f : 𝕜 → E has power series representation p on a ball of some radius and for each positive radius it has some power series representation, then p converges to f on the whole 𝕜.

### 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 $$f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k = \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.$$ The corresponding power series has thus a k-th coefficient equal to $\sum_{n} \binom{n}{k} p_n y^{n-k}$. 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.

noncomputable def formal_multilinear_series.change_origin_series_term {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (s : finset (fin (k + l))) (hs : s.card = l) :
(λ (i : fin l), E) (λ (i : fin k), E) F)

A term of formal_multilinear_series.change_origin_series.

Given a formal multilinear series p and a point x in its ball of convergence, p.change_origin x is a formal multilinear series such that p.sum (x+y) = (p.change_origin x).sum y when this makes sense. Each term of p.change_origin x is itself an analytic function of x given by the series p.change_origin_series. Each term in change_origin_series is the sum of change_origin_series_term's over all s of cardinality l. The definition is such that p.change_origin_series_term k l s hs (λ _, x) (λ _, y) = p (k + l) (s.piecewise (λ _, x) (λ _, y))

Equations
• s hs = _) (p (k + l))
theorem formal_multilinear_series.change_origin_series_term_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (s : finset (fin (k + l))) (hs : s.card = l) (x y : E) :
( s hs) (λ (_x : fin l), x)) (λ (_x : fin k), y) = (p (k + l)) (s.piecewise (λ (_x : fin (k + l)), x) (λ (_x : fin (k + l)), y))
@[simp]
theorem formal_multilinear_series.norm_change_origin_series_term {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (s : finset (fin (k + l))) (hs : s.card = l) :
s hs = p (k + l)
@[simp]
theorem formal_multilinear_series.nnnorm_change_origin_series_term {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (s : finset (fin (k + l))) (hs : s.card = l) :
s hs∥₊ = p (k + l)∥₊
theorem formal_multilinear_series.nnnorm_change_origin_series_term_apply_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (s : finset (fin (k + l))) (hs : s.card = l) (x y : E) :
( s hs) (λ (_x : fin l), x)) (λ (_x : fin k), y)∥₊ p (k + l)∥₊ * x∥₊ ^ l * y∥₊ ^ k
noncomputable def formal_multilinear_series.change_origin_series {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k : ) :
(λ (i : fin k), E) F)

The power series for f.change_origin k.

Given a formal multilinear series p and a point x in its ball of convergence, p.change_origin x is a formal multilinear series such that p.sum (x+y) = (p.change_origin x).sum y when this makes sense. Its k-th term is the sum of the series p.change_origin_series k.

Equations
theorem formal_multilinear_series.nnnorm_change_origin_series_le_tsum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) :
l∥₊ ∑' (x : {s // s.card = l}), p (k + l)∥₊
theorem formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k l : ) (x : E) :
l) (λ (_x : fin l), x)∥₊ ∑' (s : {s // s.card = l}), p (k + l)∥₊ * x∥₊ ^ l
noncomputable def formal_multilinear_series.change_origin {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (x : E) :

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.

Equations
@[simp]
def formal_multilinear_series.change_origin_index_equiv  :
(Σ (k l : ), {s // s.card = l}) Σ (n : ), finset (fin n)

An auxiliary equivalence useful in the proofs about formal_multilinear_series.change_origin_series: the set of triples (k, l, s), where s is a finset (fin (k + l)) of cardinality l is equivalent to the set of pairs (n, s), where s is a finset (fin n).

The forward map sends (k, l, s) to (k + l, s) and the inverse map sends (n, s) to (n - finset.card s, finset.card s, s). The actual definition is less readable because of problems with non-definitional equalities.

Equations
@[simp]
theorem formal_multilinear_series.change_origin_series_summable_aux₁ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r r' : nnreal} (hr : r + r' < p.radius) :
summable (λ (s : Σ (k l : ), {s // s.card = l}), p (s.fst + s.snd.fst)∥₊ * r ^ s.snd.fst * r' ^ s.fst)
theorem formal_multilinear_series.change_origin_series_summable_aux₂ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (hr : r < p.radius) (k : ) :
summable (λ (s : Σ (l : ), {s // s.card = l}), p (k + s.fst)∥₊ * r ^ s.fst)
theorem formal_multilinear_series.change_origin_series_summable_aux₃ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {r : nnreal} (hr : r < p.radius) (k : ) :
summable (λ (l : ), l∥₊ * r ^ l)
theorem formal_multilinear_series.le_change_origin_series_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k : ) :
theorem formal_multilinear_series.nnnorm_change_origin_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} (k : ) (h : x∥₊ < p.radius) :
k∥₊ ∑' (s : Σ (l : ), {s // s.card = l}), p (k + s.fst)∥₊ * x∥₊ ^ s.fst
theorem formal_multilinear_series.change_origin_radius {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x : E} :

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.

theorem formal_multilinear_series.has_fpower_series_on_ball_change_origin {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) (k : ) (hr : 0 < p.radius) :
theorem formal_multilinear_series.change_origin_eval {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] (p : F) {x y : E} (h : < p.radius) :
(p.change_origin x).sum y = p.sum (x + y)

Summing the series p.change_origin x at a point y gives back p (x + y)

theorem has_fpower_series_on_ball.change_origin {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x y : E} {r : ennreal} (hf : r) (h : y∥₊ < r) :
(x + y) (r - 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.

theorem has_fpower_series_on_ball.analytic_at_of_mem {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x y : E} {r : ennreal} (hf : r) (h : y r) :
f y

If a function admits a power series expansion p on an open ball B (x, r), then it is analytic at every point of this ball.

theorem has_fpower_series_on_ball.analytic_on {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ E] [ F] {f : E → F} {p : F} {x : E} {r : ennreal} (hf : r) :
f r)
theorem is_open_analytic_at (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [ E] [ F] (f : E → F) :
is_open {x : E | f x}

For any function f from a normed vector space to a Banach space, the set of points x such that f is analytic at x is open.