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analysis.special_functions.exponential

Calculus results on exponential in a Banach algebra #

In this file, we prove basic properties about the derivative of the exponential map exp 𝕂 in a Banach algebra 𝔸 over a field 𝕂. We keep them separate from the main file analysis/normed_space/exponential in order to minimize dependencies.

Main results #

We prove most result for an arbitrary field 𝕂, and then specialize to 𝕂 = ℝ or 𝕂 = β„‚.

General case #

𝕂 = ℝ or 𝕂 = β„‚ #

Compatibilty with real.exp and complex.exp #

theorem has_strict_fderiv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [nondiscrete_normed_field 𝕂] [normed_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂) 1 0

The exponential in a Banach-algebra 𝔸 over a normed field 𝕂 has strict FrΓ©chet-derivative 1 : 𝔸 β†’L[𝕂] 𝔸 at zero, as long as it converges on a neighborhood of zero.

theorem has_fderiv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [nondiscrete_normed_field 𝕂] [normed_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (exp 𝕂) 1 0

The exponential in a Banach-algebra 𝔸 over a normed field 𝕂 has FrΓ©chet-derivative 1 : 𝔸 β†’L[𝕂] 𝔸 at zero, as long as it converges on a neighborhood of zero.

theorem has_fderiv_at_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [nondiscrete_normed_field 𝕂] [normed_comm_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] [char_zero 𝕂] {x : 𝔸} (hx : x ∈ emetric.ball 0 (exp_series 𝕂 𝔸).radius) :
has_fderiv_at (exp 𝕂) (exp 𝕂 x β€’ 1) x

The exponential map in a commutative Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero has FrΓ©chet-derivative exp 𝕂 x β€’ 1 : 𝔸 β†’L[𝕂] 𝔸 at any point x in the disk of convergence.

theorem has_strict_fderiv_at_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [nondiscrete_normed_field 𝕂] [normed_comm_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] [char_zero 𝕂] {x : 𝔸} (hx : x ∈ emetric.ball 0 (exp_series 𝕂 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂) (exp 𝕂 x β€’ 1) x

The exponential map in a commutative Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero has strict FrΓ©chet-derivative exp 𝕂 x β€’ 1 : 𝔸 β†’L[𝕂] 𝔸 at any point x in the disk of convergence.

theorem has_strict_deriv_at_exp_of_mem_ball {𝕂 : Type u_1} [nondiscrete_normed_field 𝕂] [complete_space 𝕂] [char_zero 𝕂] {x : 𝕂} (hx : x ∈ emetric.ball 0 (exp_series 𝕂 𝕂).radius) :
has_strict_deriv_at (exp 𝕂) (exp 𝕂 x) x

The exponential map in a complete normed field 𝕂 of characteristic zero has strict derivative exp 𝕂 x at any point x in the disk of convergence.

theorem has_deriv_at_exp_of_mem_ball {𝕂 : Type u_1} [nondiscrete_normed_field 𝕂] [complete_space 𝕂] [char_zero 𝕂] {x : 𝕂} (hx : x ∈ emetric.ball 0 (exp_series 𝕂 𝕂).radius) :
has_deriv_at (exp 𝕂) (exp 𝕂 x) x

The exponential map in a complete normed field 𝕂 of characteristic zero has derivative exp 𝕂 x at any point x in the disk of convergence.

theorem has_strict_deriv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} [nondiscrete_normed_field 𝕂] [complete_space 𝕂] (h : 0 < (exp_series 𝕂 𝕂).radius) :
has_strict_deriv_at (exp 𝕂) 1 0

The exponential map in a complete normed field 𝕂 of characteristic zero has strict derivative 1 at zero, as long as it converges on a neighborhood of zero.

theorem has_deriv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} [nondiscrete_normed_field 𝕂] [complete_space 𝕂] (h : 0 < (exp_series 𝕂 𝕂).radius) :
has_deriv_at (exp 𝕂) 1 0

The exponential map in a complete normed field 𝕂 of characteristic zero has derivative 1 at zero, as long as it converges on a neighborhood of zero.

theorem has_strict_fderiv_at_exp_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] :
has_strict_fderiv_at (exp 𝕂) 1 0

The exponential in a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = β„‚ has strict FrΓ©chet-derivative 1 : 𝔸 β†’L[𝕂] 𝔸 at zero.

theorem has_fderiv_at_exp_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] :
has_fderiv_at (exp 𝕂) 1 0

The exponential in a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = β„‚ has FrΓ©chet-derivative 1 : 𝔸 β†’L[𝕂] 𝔸 at zero.

theorem has_strict_fderiv_at_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_comm_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] {x : 𝔸} :
has_strict_fderiv_at (exp 𝕂) (exp 𝕂 x β€’ 1) x

The exponential map in a commutative Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = β„‚ has strict FrΓ©chet-derivative exp 𝕂 x β€’ 1 : 𝔸 β†’L[𝕂] 𝔸 at any point x.

theorem has_fderiv_at_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_comm_ring 𝔸] [normed_algebra 𝕂 𝔸] [complete_space 𝔸] {x : 𝔸} :
has_fderiv_at (exp 𝕂) (exp 𝕂 x β€’ 1) x

The exponential map in a commutative Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = β„‚ has FrΓ©chet-derivative exp 𝕂 x β€’ 1 : 𝔸 β†’L[𝕂] 𝔸 at any point x.

theorem has_strict_deriv_at_exp {𝕂 : Type u_1} [is_R_or_C 𝕂] {x : 𝕂} :
has_strict_deriv_at (exp 𝕂) (exp 𝕂 x) x

The exponential map in 𝕂 = ℝ or 𝕂 = β„‚ has strict derivative exp 𝕂 x at any point x.

theorem has_deriv_at_exp {𝕂 : Type u_1} [is_R_or_C 𝕂] {x : 𝕂} :
has_deriv_at (exp 𝕂) (exp 𝕂 x) x

The exponential map in 𝕂 = ℝ or 𝕂 = β„‚ has derivative exp 𝕂 x at any point x.

theorem has_strict_deriv_at_exp_zero {𝕂 : Type u_1} [is_R_or_C 𝕂] :
has_strict_deriv_at (exp 𝕂) 1 0

The exponential map in 𝕂 = ℝ or 𝕂 = β„‚ has strict derivative 1 at zero.

theorem has_deriv_at_exp_zero {𝕂 : Type u_1} [is_R_or_C 𝕂] :
has_deriv_at (exp 𝕂) 1 0

The exponential map in 𝕂 = ℝ or 𝕂 = β„‚ has derivative 1 at zero.