Exponential, trigonometric and hyperbolic trigonometric functions

This file contains the definitions of the real and complex exponential, sine, cosine, tangent, hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.

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

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

def Complex.exp' (z : ℂ) : CauSeq ℂ ⇑abs

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

Equations

• Complex.exp' z = ⟨fun (n : ℕ) => ∑ m ∈ Finset.range n, z ^ m / ↑m.factorial, ⋯⟩

def Complex.exp (z : ℂ) : ℂ

The complex exponential function, defined via its Taylor series

Equations

• Complex.exp z = (Complex.exp' z).lim

def Complex.sin (z : ℂ) : ℂ

The complex sine function, defined via exp

Equations

• Complex.sin z = (Complex.exp (-z * Complex.I) - Complex.exp (z * Complex.I)) * Complex.I / 2

def Complex.cos (z : ℂ) : ℂ

The complex cosine function, defined via exp

Equations

• Complex.cos z = (Complex.exp (z * Complex.I) + Complex.exp (-z * Complex.I)) / 2

def Complex.tan (z : ℂ) : ℂ

The complex tangent function, defined as sin z / cos z

Equations

• Complex.tan z = Complex.sin z / Complex.cos z

def Complex.cot (z : ℂ) : ℂ

The complex cotangent function, defined as cos z / sin z

Equations

• z.cot = Complex.cos z / Complex.sin z

def Complex.sinh (z : ℂ) : ℂ

The complex hyperbolic sine function, defined via exp

Equations

• Complex.sinh z = (Complex.exp z - Complex.exp (-z)) / 2

def Complex.cosh (z : ℂ) : ℂ

The complex hyperbolic cosine function, defined via exp

Equations

• Complex.cosh z = (Complex.exp z + Complex.exp (-z)) / 2

def Complex.tanh (z : ℂ) : ℂ

The complex hyperbolic tangent function, defined as sinh z / cosh z

Equations

• Complex.tanh z = Complex.sinh z / Complex.cosh z

def Complex.termCexp : Lean.ParserDescr

scoped notation for the complex exponential function

Equations

• Complex.termCexp = Lean.ParserDescr.node `Complex.termCexp 1024 (Lean.ParserDescr.symbol "cexp")

def Real.exp (x : ℝ) : ℝ

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

Equations

• Real.exp x = (Complex.exp ↑x).re

def Real.sin (x : ℝ) : ℝ

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

Equations

• Real.sin x = (Complex.sin ↑x).re

def Real.cos (x : ℝ) : ℝ

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

Equations

• Real.cos x = (Complex.cos ↑x).re

def Real.tan (x : ℝ) : ℝ

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

Equations

• Real.tan x = (Complex.tan ↑x).re

def Real.cot (x : ℝ) : ℝ

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

Equations

• x.cot = (↑x).cot.re

def Real.sinh (x : ℝ) : ℝ

The real hypebolic sine function, defined as the real part of the complex hyperbolic sine

Equations

• Real.sinh x = (Complex.sinh ↑x).re

def Real.cosh (x : ℝ) : ℝ

The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine

Equations

• Real.cosh x = (Complex.cosh ↑x).re

def Real.tanh (x : ℝ) : ℝ

The real hypebolic tangent function, defined as the real part of the complex hyperbolic tangent

Equations

• Real.tanh x = (Complex.tanh ↑x).re

def Real.termRexp : Lean.ParserDescr

scoped notation for the real exponential function

Equations

• Real.termRexp = Lean.ParserDescr.node `Real.termRexp 1024 (Lean.ParserDescr.symbol "rexp")

@[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

Equations

• Complex.expMonoidHom = { toFun := fun (z : Multiplicative ℂ) => Complex.exp (Multiplicative.toAdd z), map_one' := Complex.expMonoidHom.proof_1, map_mul' := Complex.expMonoidHom.proof_2 }

@[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_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

theorem Complex.two_sinh (x : ℂ) : 2 * sinh x = exp x - exp (-x)

theorem Complex.two_cosh (x : ℂ) : 2 * cosh x = exp x + exp (-x)

@[simp]

theorem Complex.sinh_zero : sinh 0 = 0

@[simp]

theorem Complex.sinh_neg (x : ℂ) : sinh (-x) = -sinh x

theorem Complex.sinh_add (x y : ℂ) : sinh (x + y) = sinh x * cosh y + cosh x * sinh y

@[simp]

theorem Complex.cosh_zero : cosh 0 = 1

@[simp]

theorem Complex.cosh_neg (x : ℂ) : cosh (-x) = cosh x

theorem Complex.cosh_add (x y : ℂ) : cosh (x + y) = cosh x * cosh y + sinh x * sinh y

theorem Complex.sinh_sub (x y : ℂ) : sinh (x - y) = sinh x * cosh y - cosh x * sinh y

theorem Complex.cosh_sub (x y : ℂ) : cosh (x - y) = cosh x * cosh y - sinh x * sinh y

theorem Complex.sinh_conj (x : ℂ) : sinh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (sinh x)

@[simp]

theorem Complex.ofReal_sinh_ofReal_re (x : ℝ) : ↑(sinh ↑x).re = sinh ↑x

@[simp]

theorem Complex.ofReal_sinh (x : ℝ) : ↑(Real.sinh x) = sinh ↑x

@[simp]

theorem Complex.sinh_ofReal_im (x : ℝ) : (sinh ↑x).im = 0

theorem Complex.sinh_ofReal_re (x : ℝ) : (sinh ↑x).re = Real.sinh x

theorem Complex.cosh_conj (x : ℂ) : cosh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (cosh x)

theorem Complex.ofReal_cosh_ofReal_re (x : ℝ) : ↑(cosh ↑x).re = cosh ↑x

@[simp]

theorem Complex.ofReal_cosh (x : ℝ) : ↑(Real.cosh x) = cosh ↑x

@[simp]

theorem Complex.cosh_ofReal_im (x : ℝ) : (cosh ↑x).im = 0

@[simp]

theorem Complex.cosh_ofReal_re (x : ℝ) : (cosh ↑x).re = Real.cosh x

theorem Complex.tanh_eq_sinh_div_cosh (x : ℂ) : tanh x = sinh x / cosh x

@[simp]

theorem Complex.tanh_zero : tanh 0 = 0

@[simp]

theorem Complex.tanh_neg (x : ℂ) : tanh (-x) = -tanh x

theorem Complex.tanh_conj (x : ℂ) : tanh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (tanh x)

@[simp]

theorem Complex.ofReal_tanh_ofReal_re (x : ℝ) : ↑(tanh ↑x).re = tanh ↑x

@[simp]

theorem Complex.ofReal_tanh (x : ℝ) : ↑(Real.tanh x) = tanh ↑x

@[simp]

theorem Complex.tanh_ofReal_im (x : ℝ) : (tanh ↑x).im = 0

theorem Complex.tanh_ofReal_re (x : ℝ) : (tanh ↑x).re = Real.tanh x

@[simp]

theorem Complex.cosh_add_sinh (x : ℂ) : cosh x + sinh x = exp x

@[simp]

theorem Complex.sinh_add_cosh (x : ℂ) : sinh x + cosh x = exp x

@[simp]

theorem Complex.exp_sub_cosh (x : ℂ) : exp x - cosh x = sinh x

@[simp]

theorem Complex.exp_sub_sinh (x : ℂ) : exp x - sinh x = cosh x

@[simp]

theorem Complex.cosh_sub_sinh (x : ℂ) : cosh x - sinh x = exp (-x)

@[simp]

theorem Complex.sinh_sub_cosh (x : ℂ) : sinh x - cosh x = -exp (-x)

@[simp]

theorem Complex.cosh_sq_sub_sinh_sq (x : ℂ) : cosh x ^ 2 - sinh x ^ 2 = 1

theorem Complex.cosh_sq (x : ℂ) : cosh x ^ 2 = sinh x ^ 2 + 1

theorem Complex.sinh_sq (x : ℂ) : sinh x ^ 2 = cosh x ^ 2 - 1

theorem Complex.cosh_two_mul (x : ℂ) : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2

theorem Complex.sinh_two_mul (x : ℂ) : sinh (2 * x) = 2 * sinh x * cosh x

theorem Complex.cosh_three_mul (x : ℂ) : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x

theorem Complex.sinh_three_mul (x : ℂ) : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x

@[simp]

theorem Complex.sin_zero : sin 0 = 0

@[simp]

theorem Complex.sin_neg (x : ℂ) : sin (-x) = -sin x

theorem Complex.two_sin (x : ℂ) : 2 * sin x = (exp (-x * I) - exp (x * I)) * I

theorem Complex.two_cos (x : ℂ) : 2 * cos x = exp (x * I) + exp (-x * I)

theorem Complex.sinh_mul_I (x : ℂ) : sinh (x * I) = sin x * I

theorem Complex.cosh_mul_I (x : ℂ) : cosh (x * I) = cos x

theorem Complex.tanh_mul_I (x : ℂ) : tanh (x * I) = tan x * I

theorem Complex.cos_mul_I (x : ℂ) : cos (x * I) = cosh x

theorem Complex.sin_mul_I (x : ℂ) : sin (x * I) = sinh x * I

theorem Complex.tan_mul_I (x : ℂ) : tan (x * I) = tanh x * I

theorem Complex.sin_add (x y : ℂ) : sin (x + y) = sin x * cos y + cos x * sin y

@[simp]

theorem Complex.cos_zero : cos 0 = 1

@[simp]

theorem Complex.cos_neg (x : ℂ) : cos (-x) = cos x

theorem Complex.cos_add (x y : ℂ) : cos (x + y) = cos x * cos y - sin x * sin y

theorem Complex.sin_sub (x y : ℂ) : sin (x - y) = sin x * cos y - cos x * sin y

theorem Complex.cos_sub (x y : ℂ) : cos (x - y) = cos x * cos y + sin x * sin y

theorem Complex.sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I

theorem Complex.sin_eq (z : ℂ) : sin z = sin ↑z.re * cosh ↑z.im + cos ↑z.re * sinh ↑z.im * I

theorem Complex.cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I

theorem Complex.cos_eq (z : ℂ) : cos z = cos ↑z.re * cosh ↑z.im - sin ↑z.re * sinh ↑z.im * I

theorem Complex.sin_sub_sin (x y : ℂ) : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)

theorem Complex.cos_sub_cos (x y : ℂ) : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2)

theorem Complex.sin_add_sin (x y : ℂ) : sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2)

theorem Complex.cos_add_cos (x y : ℂ) : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2)

theorem Complex.sin_conj (x : ℂ) : sin ((starRingEnd ℂ) x) = (starRingEnd ℂ) (sin x)

@[simp]

theorem Complex.ofReal_sin_ofReal_re (x : ℝ) : ↑(sin ↑x).re = sin ↑x

@[simp]

theorem Complex.ofReal_sin (x : ℝ) : ↑(Real.sin x) = sin ↑x

@[simp]

theorem Complex.sin_ofReal_im (x : ℝ) : (sin ↑x).im = 0

theorem Complex.sin_ofReal_re (x : ℝ) : (sin ↑x).re = Real.sin x

theorem Complex.cos_conj (x : ℂ) : cos ((starRingEnd ℂ) x) = (starRingEnd ℂ) (cos x)

@[simp]

theorem Complex.ofReal_cos_ofReal_re (x : ℝ) : ↑(cos ↑x).re = cos ↑x

@[simp]

theorem Complex.ofReal_cos (x : ℝ) : ↑(Real.cos x) = cos ↑x

@[simp]

theorem Complex.cos_ofReal_im (x : ℝ) : (cos ↑x).im = 0

theorem Complex.cos_ofReal_re (x : ℝ) : (cos ↑x).re = Real.cos x

@[simp]

theorem Complex.tan_zero : tan 0 = 0

theorem Complex.tan_eq_sin_div_cos (x : ℂ) : tan x = sin x / cos x

theorem Complex.cot_eq_cos_div_sin (x : ℂ) : x.cot = cos x / sin x

theorem Complex.tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x

@[simp]

theorem Complex.tan_neg (x : ℂ) : tan (-x) = -tan x

theorem Complex.tan_conj (x : ℂ) : tan ((starRingEnd ℂ) x) = (starRingEnd ℂ) (tan x)

theorem Complex.cot_conj (x : ℂ) : ((starRingEnd ℂ) x).cot = (starRingEnd ℂ) x.cot

@[simp]

theorem Complex.ofReal_tan_ofReal_re (x : ℝ) : ↑(tan ↑x).re = tan ↑x

@[simp]

theorem Complex.ofReal_cot_ofReal_re (x : ℝ) : ↑(↑x).cot.re = (↑x).cot

@[simp]

theorem Complex.ofReal_tan (x : ℝ) : ↑(Real.tan x) = tan ↑x

@[simp]

theorem Complex.ofReal_cot (x : ℝ) : ↑x.cot = (↑x).cot

@[simp]

theorem Complex.tan_ofReal_im (x : ℝ) : (tan ↑x).im = 0

theorem Complex.tan_ofReal_re (x : ℝ) : (tan ↑x).re = Real.tan x

theorem Complex.cos_add_sin_I (x : ℂ) : cos x + sin x * I = exp (x * I)

theorem Complex.cos_sub_sin_I (x : ℂ) : cos x - sin x * I = exp (-x * I)

@[simp]

theorem Complex.sin_sq_add_cos_sq (x : ℂ) : sin x ^ 2 + cos x ^ 2 = 1

@[simp]

theorem Complex.cos_sq_add_sin_sq (x : ℂ) : cos x ^ 2 + sin x ^ 2 = 1

theorem Complex.cos_two_mul' (x : ℂ) : cos (2 * x) = cos x ^ 2 - sin x ^ 2

theorem Complex.cos_two_mul (x : ℂ) : cos (2 * x) = 2 * cos x ^ 2 - 1

theorem Complex.sin_two_mul (x : ℂ) : sin (2 * x) = 2 * sin x * cos x

theorem Complex.cos_sq (x : ℂ) : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2

theorem Complex.cos_sq' (x : ℂ) : cos x ^ 2 = 1 - sin x ^ 2

theorem Complex.sin_sq (x : ℂ) : sin x ^ 2 = 1 - cos x ^ 2

theorem Complex.inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2

theorem Complex.tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2

theorem Complex.cos_three_mul (x : ℂ) : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x

theorem Complex.sin_three_mul (x : ℂ) : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3

theorem Complex.exp_mul_I (x : ℂ) : exp (x * I) = cos x + sin x * I

theorem Complex.exp_add_mul_I (x y : ℂ) : exp (x + y * I) = exp x * (cos y + sin y * I)

theorem Complex.exp_eq_exp_re_mul_sin_add_cos (x : ℂ) : exp x = exp ↑x.re * (cos ↑x.im + sin ↑x.im * I)

theorem Complex.exp_re (x : ℂ) : (exp x).re = Real.exp x.re * Real.cos x.im

theorem Complex.exp_im (x : ℂ) : (exp x).im = Real.exp x.re * Real.sin x.im

@[simp]

theorem Complex.exp_ofReal_mul_I_re (x : ℝ) : (exp (↑x * I)).re = Real.cos x

@[simp]

theorem Complex.exp_ofReal_mul_I_im (x : ℝ) : (exp (↑x * I)).im = Real.sin x

theorem Complex.cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) : (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I

De Moivre's formula

@[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

Equations

• Real.expMonoidHom = { toFun := fun (x : Multiplicative ℝ) => Real.exp (Multiplicative.toAdd x), map_one' := Real.expMonoidHom.proof_1, map_mul' := Real.expMonoidHom.proof_2 }

@[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

@[simp]

theorem Real.sin_zero : sin 0 = 0

@[simp]

theorem Real.sin_neg (x : ℝ) : sin (-x) = -sin x

theorem Real.sin_add (x y : ℝ) : sin (x + y) = sin x * cos y + cos x * sin y

@[simp]

theorem Real.cos_zero : cos 0 = 1

@[simp]

theorem Real.cos_neg (x : ℝ) : cos (-x) = cos x

@[simp]

theorem Real.cos_abs (x : ℝ) : cos |x| = cos x

theorem Real.cos_add (x y : ℝ) : cos (x + y) = cos x * cos y - sin x * sin y

theorem Real.sin_sub (x y : ℝ) : sin (x - y) = sin x * cos y - cos x * sin y

theorem Real.cos_sub (x y : ℝ) : cos (x - y) = cos x * cos y + sin x * sin y

theorem Real.sin_sub_sin (x y : ℝ) : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)

theorem Real.cos_sub_cos (x y : ℝ) : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2)

theorem Real.cos_add_cos (x y : ℝ) : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2)

theorem Real.two_mul_sin_mul_sin (x y : ℝ) : 2 * sin x * sin y = cos (x - y) - cos (x + y)

theorem Real.two_mul_cos_mul_cos (x y : ℝ) : 2 * cos x * cos y = cos (x - y) + cos (x + y)

theorem Real.two_mul_sin_mul_cos (x y : ℝ) : 2 * sin x * cos y = sin (x - y) + sin (x + y)

theorem Real.tan_eq_sin_div_cos (x : ℝ) : tan x = sin x / cos x

theorem Real.cot_eq_cos_div_sin (x : ℝ) : x.cot = cos x / sin x

theorem Real.tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x

@[simp]

theorem Real.tan_zero : tan 0 = 0

@[simp]

theorem Real.tan_neg (x : ℝ) : tan (-x) = -tan x

@[simp]

theorem Real.sin_sq_add_cos_sq (x : ℝ) : sin x ^ 2 + cos x ^ 2 = 1

@[simp]

theorem Real.cos_sq_add_sin_sq (x : ℝ) : cos x ^ 2 + sin x ^ 2 = 1

theorem Real.sin_sq_le_one (x : ℝ) : sin x ^ 2 ≤ 1

theorem Real.cos_sq_le_one (x : ℝ) : cos x ^ 2 ≤ 1

theorem Real.abs_sin_le_one (x : ℝ) : |sin x| ≤ 1

theorem Real.abs_cos_le_one (x : ℝ) : |cos x| ≤ 1

theorem Real.sin_le_one (x : ℝ) : sin x ≤ 1

theorem Real.cos_le_one (x : ℝ) : cos x ≤ 1

theorem Real.neg_one_le_sin (x : ℝ) : -1 ≤ sin x

theorem Real.neg_one_le_cos (x : ℝ) : -1 ≤ cos x

theorem Real.cos_two_mul (x : ℝ) : cos (2 * x) = 2 * cos x ^ 2 - 1

theorem Real.cos_two_mul' (x : ℝ) : cos (2 * x) = cos x ^ 2 - sin x ^ 2

theorem Real.sin_two_mul (x : ℝ) : sin (2 * x) = 2 * sin x * cos x

theorem Real.cos_sq (x : ℝ) : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2

theorem Real.cos_sq' (x : ℝ) : cos x ^ 2 = 1 - sin x ^ 2

theorem Real.sin_sq (x : ℝ) : sin x ^ 2 = 1 - cos x ^ 2

theorem Real.sin_sq_eq_half_sub (x : ℝ) : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2

theorem Real.abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = √(1 - cos x ^ 2)

theorem Real.abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = √(1 - sin x ^ 2)

theorem Real.inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2

theorem Real.tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2

theorem Real.inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (√(1 + tan x ^ 2))⁻¹ = cos x

theorem Real.tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : tan x / √(1 + tan x ^ 2) = sin x

theorem Real.cos_three_mul (x : ℝ) : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x

theorem Real.sin_three_mul (x : ℝ) : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3

theorem Real.sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2

The definition of sinh in terms of exp.

@[simp]

theorem Real.sinh_zero : sinh 0 = 0

@[simp]

theorem Real.sinh_neg (x : ℝ) : sinh (-x) = -sinh x

theorem Real.sinh_add (x y : ℝ) : sinh (x + y) = sinh x * cosh y + cosh x * sinh y

theorem Real.cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2

The definition of cosh in terms of exp.

@[simp]

theorem Real.cosh_zero : cosh 0 = 1

@[simp]

theorem Real.cosh_neg (x : ℝ) : cosh (-x) = cosh x

@[simp]

theorem Real.cosh_abs (x : ℝ) : cosh |x| = cosh x

theorem Real.cosh_add (x y : ℝ) : cosh (x + y) = cosh x * cosh y + sinh x * sinh y

theorem Real.sinh_sub (x y : ℝ) : sinh (x - y) = sinh x * cosh y - cosh x * sinh y

theorem Real.cosh_sub (x y : ℝ) : cosh (x - y) = cosh x * cosh y - sinh x * sinh y

theorem Real.tanh_eq_sinh_div_cosh (x : ℝ) : tanh x = sinh x / cosh x

@[simp]

theorem Real.tanh_zero : tanh 0 = 0

@[simp]

theorem Real.tanh_neg (x : ℝ) : tanh (-x) = -tanh x

@[simp]

theorem Real.cosh_add_sinh (x : ℝ) : cosh x + sinh x = exp x

@[simp]

theorem Real.sinh_add_cosh (x : ℝ) : sinh x + cosh x = exp x

@[simp]

theorem Real.exp_sub_cosh (x : ℝ) : exp x - cosh x = sinh x

@[simp]

theorem Real.exp_sub_sinh (x : ℝ) : exp x - sinh x = cosh x

@[simp]

theorem Real.cosh_sub_sinh (x : ℝ) : cosh x - sinh x = exp (-x)

@[simp]

theorem Real.sinh_sub_cosh (x : ℝ) : sinh x - cosh x = -exp (-x)

@[simp]

theorem Real.cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1

theorem Real.cosh_sq (x : ℝ) : cosh x ^ 2 = sinh x ^ 2 + 1

theorem Real.cosh_sq' (x : ℝ) : cosh x ^ 2 = 1 + sinh x ^ 2

theorem Real.sinh_sq (x : ℝ) : sinh x ^ 2 = cosh x ^ 2 - 1

theorem Real.cosh_two_mul (x : ℝ) : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2

theorem Real.sinh_two_mul (x : ℝ) : sinh (2 * x) = 2 * sinh x * cosh x

theorem Real.cosh_three_mul (x : ℝ) : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x

theorem Real.sinh_three_mul (x : ℝ) : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x

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

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 Real.cosh_pos (x : ℝ) : 0 < cosh x

Real.cosh is always positive

theorem Real.sinh_lt_cosh (x : ℝ) : sinh x < cosh x

theorem Complex.sum_div_factorial_le {α : Type u_1} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) : ∑ m ∈ Finset.filter (fun (m : ℕ) => n ≤ m) (Finset.range j), 1 / ↑m.factorial ≤ ↑n.succ / (↑n.factorial * ↑n)

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

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

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

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

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

theorem Complex.abs_exp_le_exp_abs (x : ℂ) : abs (exp x) ≤ Real.exp (abs x)

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

Equations

• Real.expNear n x r = ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial + x ^ n / ↑n.factorial * 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.cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96)

theorem Real.sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96)

theorem Real.cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x

theorem Real.sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x

theorem Real.sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x

theorem Real.cos_one_le : cos 1 ≤ 2 / 3

theorem Real.cos_one_pos : 0 < cos 1

theorem Real.cos_two_neg : cos 2 < 0

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.

def Mathlib.Meta.Positivity.evalCosh : PositivityExt

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

@[simp]

theorem Complex.abs_cos_add_sin_mul_I (x : ℝ) : abs (cos ↑x + sin ↑x * I) = 1

@[simp]

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

@[simp]

theorem Complex.abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (↑x * I)) = 1

theorem Complex.abs_exp (z : ℂ) : abs (exp z) = Real.exp z.re

theorem Complex.abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re