mathlib documentation

measure_theory.integral.torus_integral

Integral over a torus in ℂⁿ #

In this file we define the integral of a function f : ℂⁿ → E over a torus {z : ℂⁿ | ∀ i, z i ∈ metric.sphere (c i) (R i)}. In order to do this, we define torus_map (c : ℂⁿ) (R θ : ℝⁿ) to be the point in ℂⁿ given by $z_k=c_k+R_ke^{θ_ki}$, where $i$ is the imaginary unit, then define torus_integral f c R as the integral over the cube $[0, (λ _, 2π)] = \{θ\|∀ k, 0 ≤ θ_k ≤ 2π\}$ of the Jacobian of the torus_map multiplied by f (torus_map c R θ).

We also define a predicate saying that f ∘ torus_map c R is integrable on the cube [0, (λ _, 2\pi)].

Main definitions #

Main statements #

Notations #

Tags #

integral, torus

torus_map, a generalization of a torus #

noncomputable def torus_map {n : } (c : fin n) (R : fin n) :
(fin n)fin n

The n dimensional exponential map $θ_i ↦ c + R e^{θ_i*I}, θ ∈ ℝⁿ$ representing a torus in ℂⁿ with center c ∈ ℂⁿ and generalized radius R ∈ ℝⁿ, so we can adjust it to every n axis.

Equations
theorem torus_map_sub_center {n : } (c : fin n) (R θ : fin n) :
torus_map c R θ - c = torus_map 0 R θ
theorem torus_map_eq_center_iff {n : } {c : fin n} {R θ : fin n} :
torus_map c R θ = c R = 0
@[simp]
theorem torus_map_zero_radius {n : } (c : fin n) :

Integrability of a function on a generalized torus #

def torus_integrable {n : } {E : Type u_1} [normed_group E] (f : (fin n) → E) (c : fin n) (R : fin n) :
Prop

A function f : ℂⁿ → E is integrable on the generalized torus if the function f ∘ torus_map c R θ is integrable on Icc (0 : ℝⁿ) (λ _, 2 * π)

Equations
theorem torus_integrable.torus_integrable_const {n : } {E : Type u_1} [normed_group E] (a : E) (c : fin n) (R : fin n) :
torus_integrable (λ (_x : fin n), a) c R

Constant functions are torus integrable

@[protected]
theorem torus_integrable.neg {n : } {E : Type u_1} [normed_group E] {f : (fin n) → E} {c : fin n} {R : fin n} (hf : torus_integrable f c R) :

If f is torus integrable then -f is torus integrable.

@[protected]
theorem torus_integrable.add {n : } {E : Type u_1} [normed_group E] {f g : (fin n) → E} {c : fin n} {R : fin n} (hf : torus_integrable f c R) (hg : torus_integrable g c R) :

If f and g are two torus integrable functions, then so is f + g.

@[protected]
theorem torus_integrable.sub {n : } {E : Type u_1} [normed_group E] {f g : (fin n) → E} {c : fin n} {R : fin n} (hf : torus_integrable f c R) (hg : torus_integrable g c R) :

If f and g are two torus integrable functions, then so is f - g.

theorem torus_integrable.torus_integrable_zero_radius {n : } {E : Type u_1} [normed_group E] {f : (fin n) → E} {c : fin n} :
theorem torus_integrable.function_integrable {n : } {E : Type u_1} [normed_group E] {f : (fin n) → E} {c : fin n} {R : fin n} [normed_space E] (hf : torus_integrable f c R) :

The function given in the definition of torus_integral is integrable.

noncomputable def torus_integral {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] (f : (fin n) → E) (c : fin n) (R : fin n) :
E

The definition of the integral over a generalized torus with center c ∈ ℂⁿ and radius R ∈ ℝⁿ as the -product of the derivative of torus_map and f (torus_map c R θ)

Equations
theorem torus_integral_radius_zero {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] (hn : n 0) (f : (fin n) → E) (c : fin n) :
∯ (x : fin n) in T(c, 0), f x = 0
theorem torus_integral_neg {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] (f : (fin n) → E) (c : fin n) (R : fin n) :
∯ (x : fin n) in T(c, R), -f x = -∯ (x : fin n) in T(c, R), f x
theorem torus_integral_add {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {f g : (fin n) → E} {c : fin n} {R : fin n} (hf : torus_integrable f c R) (hg : torus_integrable g c R) :
∯ (x : fin n) in T(c, R), f x + g x = (∯ (x : fin n) in T(c, R), f x) + ∯ (x : fin n) in T(c, R), g x
theorem torus_integral_sub {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {f g : (fin n) → E} {c : fin n} {R : fin n} (hf : torus_integrable f c R) (hg : torus_integrable g c R) :
∯ (x : fin n) in T(c, R), f x - g x = (∯ (x : fin n) in T(c, R), f x) - ∯ (x : fin n) in T(c, R), g x
theorem torus_integral_smul {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {𝕜 : Type u_2} [is_R_or_C 𝕜] [normed_space 𝕜 E] [smul_comm_class 𝕜 E] (a : 𝕜) (f : (fin n) → E) (c : fin n) (R : fin n) :
∯ (x : fin n) in T(c, R), a f x = a ∯ (x : fin n) in T(c, R), f x
theorem torus_integral_const_mul {n : } (a : ) (f : (fin n)) (c : fin n) (R : fin n) :
∯ (x : fin n) in T(c, R), a * f x = a * ∯ (x : fin n) in T(c, R), f x
theorem norm_torus_integral_le_of_norm_le_const {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {f : (fin n) → E} {c : fin n} {R : fin n} {C : } (hf : ∀ (θ : fin n), f (torus_map c R θ) C) :
∯ (x : fin n) in T(c, R), f x (2 * real.pi) ^ n * finset.univ.prod (λ (i : fin n), |R i|) * C

If for all θ : ℝⁿ, ∥f (torus_map c R θ)∥ is less than or equal to a constant C : ℝ, then ∥∯ x in T(c, R), f x∥ is less than or equal to (2 * π)^n * (∏ i, |R i|) * C

@[simp]
theorem torus_integral_dim0 {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] (f : (fin 0) → E) (c : fin 0) (R : fin 0) :
∯ (x : fin 0) in T(c, R), f x = f c
theorem torus_integral_dim1 {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] (f : (fin 1) → E) (c : fin 1) (R : fin 1) :
∯ (x : fin 1) in T(c, R), f x = ∮ (z : ) in C(c 0, R 0), f (λ (_x : fin 1), z)

In dimension one, torus_integral is the same as circle_integral (up to the natural equivalence between and fin 1 → ℂ).

theorem torus_integral_succ_above {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {f : (fin (n + 1)) → E} {c : fin (n + 1)} {R : fin (n + 1)} (hf : torus_integrable f c R) (i : fin (n + 1)) :
∯ (x : fin (n + 1)) in T(c, R), f x = ∮ (x : ) in C(c i, R i), ∯ (y : fin n) in T(c (i.succ_above), R (i.succ_above)), f (i.insert_nth x y)

Recurrent formula for torus_integral, see also torus_integral_succ.

theorem torus_integral_succ {n : } {E : Type u_1} [normed_group E] [normed_space E] [complete_space E] {f : (fin (n + 1)) → E} {c : fin (n + 1)} {R : fin (n + 1)} (hf : torus_integrable f c R) :
∯ (x : fin (n + 1)) in T(c, R), f x = ∮ (x : ) in C(c 0, R 0), ∯ (y : fin n) in T(c fin.succ, R fin.succ), f (fin.cons x y)

Recurrent formula for torus_integral, see also torus_integral_succ_above.