Exponential Function

This file contains the definitions of the real and complex exponential function.

Main definitions

• Complex.exp: The complex exponential function, defined via its Taylor series

• Real.exp: The real exponential function, defined as the real part of the complex exponential

theorem Complex.isCauSeq_norm_exp (z : ℂ) : IsCauSeq abs fun (n : ℕ) => ∑ m ∈ Finset.range n, ‖z ^ m / ↑m.factorial‖

theorem Complex.isCauSeq_exp (z : ℂ) : IsCauSeq (fun (x : ℂ) => ‖x‖) fun (n : ℕ) => ∑ m ∈ Finset.range n, z ^ m / ↑m.factorial

def Complex.exp' (z : ℂ) : CauSeq ℂ fun (x : ℂ) => ‖x‖

The Cauchy sequence consisting of partial sums of the Taylor series of the complex exponential function

theorem Complex.exp_def (z : ℂ) : exp z = (exp' z).lim

@[irreducible]

def Complex.exp (z : ℂ) : ℂ

The complex exponential function, defined via its Taylor series

def Complex.termCexp : Lean.ParserDescr

scoped notation for the complex exponential function

def Real.exp (x : ℝ) : ℝ

The real exponential function, defined as the real part of the complex exponential

def Real.termRexp : Lean.ParserDescr

scoped notation for the real exponential function

@[simp]

theorem Complex.exp_zero : exp 0 = 1

theorem Complex.exp_add (x y : ℂ) : exp (x + y) = exp x * exp y

noncomputable def Complex.expMonoidHom : Multiplicative ℂ →* ℂ

the exponential function as a monoid hom from Multiplicative ℂ to ℂ

@[simp]

theorem Complex.expMonoidHom_apply (z : Multiplicative ℂ) : expMonoidHom z = exp (Multiplicative.toAdd z)

theorem Complex.exp_list_sum (l : List ℂ) : exp l.sum = (List.map exp l).prod

theorem Complex.exp_multiset_sum (s : Multiset ℂ) : exp s.sum = (Multiset.map exp s).prod

theorem Complex.exp_sum {α : Type u_1} (s : Finset α) (f : α → ℂ) : exp (∑ x ∈ s, f x) = ∏ x ∈ s, exp (f x)

theorem Complex.exp_nsmul (x : ℂ) (n : ℕ) : exp (n • x) = exp x ^ n

theorem Complex.exp_nsmul' (x a p : ℂ) (n : ℕ) : exp (a * ↑n * x / p) = exp (a * x / p) ^ n

This is a useful version of exp_nsmul for q-expansions of modular forms.

theorem Complex.exp_nat_mul (x : ℂ) (n : ℕ) : exp (↑n * x) = exp x ^ n

@[simp]

theorem Complex.exp_ne_zero (x : ℂ) : exp x ≠ 0

theorem Complex.exp_neg (x : ℂ) : exp (-x) = (exp x)⁻¹

theorem Complex.exp_sub (x y : ℂ) : exp (x - y) = exp x / exp y

theorem Complex.exp_int_mul (z : ℂ) (n : ℤ) : exp (↑n * z) = exp z ^ n

@[simp]

theorem Complex.exp_conj (x : ℂ) : exp ((starRingEnd ℂ) x) = (starRingEnd ℂ) (exp x)

@[simp]

theorem Complex.ofReal_exp_ofReal_re (x : ℝ) : ↑(exp ↑x).re = exp ↑x

@[simp]

theorem Complex.ofReal_exp (x : ℝ) : ↑(Real.exp x) = exp ↑x

@[simp]

theorem Complex.exp_ofReal_im (x : ℝ) : (exp ↑x).im = 0

theorem Complex.exp_ofReal_re (x : ℝ) : (exp ↑x).re = Real.exp x

@[simp]

theorem Real.exp_zero : exp 0 = 1

theorem Real.exp_add (x y : ℝ) : exp (x + y) = exp x * exp y

noncomputable def Real.expMonoidHom : Multiplicative ℝ →* ℝ

the exponential function as a monoid hom from Multiplicative ℝ to ℝ

@[simp]

theorem Real.expMonoidHom_apply (x : Multiplicative ℝ) : expMonoidHom x = exp (Multiplicative.toAdd x)

theorem Real.exp_list_sum (l : List ℝ) : exp l.sum = (List.map exp l).prod

theorem Real.exp_multiset_sum (s : Multiset ℝ) : exp s.sum = (Multiset.map exp s).prod

theorem Real.exp_sum {α : Type u_1} (s : Finset α) (f : α → ℝ) : exp (∑ x ∈ s, f x) = ∏ x ∈ s, exp (f x)

theorem Real.exp_nsmul (x : ℝ) (n : ℕ) : exp (n • x) = exp x ^ n

theorem Real.exp_nat_mul (x : ℝ) (n : ℕ) : exp (↑n * x) = exp x ^ n

@[simp]

theorem Real.exp_ne_zero (x : ℝ) : exp x ≠ 0

theorem Real.exp_neg (x : ℝ) : exp (-x) = (exp x)⁻¹

theorem Real.exp_sub (x y : ℝ) : exp (x - y) = exp x / exp y

theorem Real.sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ Finset.range n, x ^ i / ↑i.factorial ≤ exp x

theorem Real.pow_div_factorial_le_exp (x : ℝ) (hx : 0 ≤ x) (n : ℕ) : x ^ n / ↑n.factorial ≤ exp x

theorem Real.quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x

theorem Real.one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x

theorem Real.exp_pos (x : ℝ) : 0 < exp x

theorem Real.exp_nonneg (x : ℝ) : 0 ≤ exp x

@[simp]

theorem Real.abs_exp (x : ℝ) : |exp x| = exp x

theorem Real.exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x)

theorem Real.exp_strictMono : StrictMono exp

@[deprecated Real.exp_strictMono (since := "2025-10-20")]

theorem Real.exp_lt_exp_of_lt {x y : ℝ} (h : x < y) : exp x < exp y

theorem Real.exp_monotone : Monotone exp

theorem Real.exp_le_exp_of_le {x y : ℝ} (h : x ≤ y) : exp x ≤ exp y

@[simp]

theorem Real.exp_lt_exp {x y : ℝ} : exp x < exp y ↔ x < y

@[simp]

theorem Real.exp_le_exp {x y : ℝ} : exp x ≤ exp y ↔ x ≤ y

theorem Real.exp_injective : Function.Injective exp

@[simp]

theorem Real.exp_eq_exp {x y : ℝ} : exp x = exp y ↔ x = y

@[simp]

theorem Real.exp_eq_one_iff (x : ℝ) : exp x = 1 ↔ x = 0

@[simp]

theorem Real.one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x

@[simp]

theorem Real.exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0

@[simp]

theorem Real.exp_le_one_iff {x : ℝ} : exp x ≤ 1 ↔ x ≤ 0

@[simp]

theorem Real.one_le_exp_iff {x : ℝ} : 1 ≤ exp x ↔ 0 ≤ x

theorem Complex.sum_div_factorial_le {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (n j : ℕ) (hn : 0 < n) : ∑ m ∈ Finset.range j with n ≤ m, 1 / ↑m.factorial ≤ ↑n.succ / (↑n.factorial * ↑n)

theorem Complex.exp_bound {x : ℂ} (hx : ‖x‖ ≤ 1) {n : ℕ} (hn : 0 < n) : ‖exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n * (↑n.succ * (↑n.factorial * ↑n)⁻¹)

theorem Complex.exp_bound' {x : ℂ} {n : ℕ} (hx : ‖x‖ / ↑n.succ ≤ 1 / 2) : ‖exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n / ↑n.factorial * 2

theorem Complex.norm_exp_sub_one_le {x : ℂ} (hx : ‖x‖ ≤ 1) : ‖exp x - 1‖ ≤ 2 * ‖x‖

theorem Complex.norm_exp_sub_one_sub_id_le {x : ℂ} (hx : ‖x‖ ≤ 1) : ‖exp x - 1 - x‖ ≤ ‖x‖ ^ 2

theorem Real.norm_exp_sub_one_sub_id_le {x : ℝ} (hx : ‖x‖ ≤ 1) : ‖exp x - 1 - x‖ ≤ ‖x‖ ^ 2

theorem Complex.norm_exp_sub_sum_le_exp_norm_sub_sum (x : ℂ) (n : ℕ) : ‖exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial‖ ≤ Real.exp ‖x‖ - ∑ m ∈ Finset.range n, ‖x‖ ^ m / ↑m.factorial

theorem Complex.norm_exp_le_exp_norm (x : ℂ) : ‖exp x‖ ≤ Real.exp ‖x‖

theorem Complex.norm_exp_sub_sum_le_norm_mul_exp (x : ℂ) (n : ℕ) : ‖exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial‖ ≤ ‖x‖ ^ n * Real.exp ‖x‖

theorem Real.exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) : |exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial| ≤ |x| ^ n * (↑n.succ / (↑n.factorial * ↑n))

theorem Real.exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) : exp x ≤ ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial + x ^ n * (↑n + 1) / (↑n.factorial * ↑n)

theorem Real.abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x|

theorem Real.abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2

noncomputable def Real.expNear (n : ℕ) (x r : ℝ) : ℝ

A finite initial segment of the exponential series, followed by an arbitrary tail. For fixed n this is just a linear map w.r.t. r, and each map is a simple linear function of the previous (see expNear_succ), with expNear n x r ⟶ exp x as n ⟶ ∞, for any r.

@[simp]

theorem Real.expNear_zero (x r : ℝ) : expNear 0 x r = r

@[simp]

theorem Real.expNear_succ (n : ℕ) (x r : ℝ) : expNear (n + 1) x r = expNear n x (1 + x / (↑n + 1) * r)

theorem Real.expNear_sub (n : ℕ) (x r₁ r₂ : ℝ) : expNear n x r₁ - expNear n x r₂ = x ^ n / ↑n.factorial * (r₁ - r₂)

theorem Real.exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) : |exp x - expNear m x 0| ≤ |x| ^ m / ↑m.factorial * ((↑m + 1) / ↑m)

theorem Real.exp_approx_succ {n : ℕ} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ) (e : |1 + x / ↑m * a₂ - a₁| ≤ b₁ - |x| / ↑m * b₂) (h : |exp x - expNear m x a₂| ≤ |x| ^ m / ↑m.factorial * b₂) : |exp x - expNear n x a₁| ≤ |x| ^ n / ↑n.factorial * b₁

theorem Real.exp_approx_end' {n : ℕ} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm) (h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) : |exp x - expNear n x a| ≤ |x| ^ n / ↑n.factorial * b

theorem Real.exp_1_approx_succ_eq {n : ℕ} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm) (h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / ↑m.factorial * (b₁ * rm)) : |exp 1 - expNear n 1 a₁| ≤ |1| ^ n / ↑n.factorial * b₁

theorem Real.exp_approx_start (x a b : ℝ) (h : |exp x - expNear 0 x a| ≤ |x| ^ 0 / ↑(Nat.factorial 0) * b) : |exp x - a| ≤ b

theorem Real.exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) : exp x < 1 / (1 - x)

theorem Real.exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) : exp x ≤ 1 / (1 - x)

theorem Real.add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < exp x

theorem Real.add_one_le_exp (x : ℝ) : x + 1 ≤ exp x

theorem Real.one_sub_lt_exp_neg {x : ℝ} (hx : x ≠ 0) : 1 - x < exp (-x)

theorem Real.one_sub_le_exp_neg (x : ℝ) : 1 - x ≤ exp (-x)

theorem Real.one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ ↑n) : (1 - t / ↑n) ^ n ≤ exp (-t)

theorem Real.le_inv_mul_exp (x : ℝ) {c : ℝ} (hc : 0 < c) : x ≤ c⁻¹ * exp (c * x)

def Mathlib.Meta.Positivity.evalExp : PositivityExt

Extension for the positivity tactic: Real.exp is always positive.

@[simp]

theorem Complex.norm_exp_ofReal (x : ℝ) : ‖exp ↑x‖ = Real.exp x