:: Exponential Function on Complex {B}anach Algebra
:: by Noboru Endou
::
:: Received April 6, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: for X being Complex_Banach_Algebra
for z being Element of X
for n being Nat holds
( z * (z #N n) = z #N (n + 1) & (z #N n) * z = z #N (n + 1) & z * (z #N n) = (z #N n) * z )

proof end;

Lm2: for X being Complex_Banach_Algebra
for n being Nat
for z, w being Element of X st z,w are_commutative holds
( w * (z #N n) = (z #N n) * w & z * (w #N n) = (w #N n) * z )

proof end;

theorem Th1: :: CLOPBAN4:1
for X being Complex_Banach_Algebra
for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent & lim (seq1 - seq2) = 0. X holds
lim seq1 = lim seq2
proof end;

theorem Th2: :: CLOPBAN4:2
for X being Complex_Banach_Algebra
for s being sequence of X
for z being Element of X st ( for n being Nat holds s . n = z ) holds
lim s = z
proof end;

theorem Th3: :: CLOPBAN4:3
for X being Complex_Banach_Algebra
for s, s9 being sequence of X st s is convergent & s9 is convergent holds
s * s9 is convergent
proof end;

theorem Th4: :: CLOPBAN4:4
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
z * s is convergent
proof end;

theorem Th5: :: CLOPBAN4:5
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
s * z is convergent
proof end;

theorem Th6: :: CLOPBAN4:6
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
lim (z * s) = z * (lim s)
proof end;

theorem Th7: :: CLOPBAN4:7
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
lim (s * z) = (lim s) * z
proof end;

theorem Th8: :: CLOPBAN4:8
for X being Complex_Banach_Algebra
for s, s9 being sequence of X st s is convergent & s9 is convergent holds
lim (s * s9) = (lim s) * (lim s9)
proof end;

theorem Th9: :: CLOPBAN4:9
for X being Complex_Banach_Algebra
for z being Element of X
for seq being sequence of X holds
( Partial_Sums (z * seq) = z * (Partial_Sums seq) & Partial_Sums (seq * z) = (Partial_Sums seq) * z )
proof end;

theorem Th10: :: CLOPBAN4:10
for X being Complex_Banach_Algebra
for k being Nat
for seq being sequence of X holds ||.((Partial_Sums seq) . k).|| <= () . k
proof end;

theorem Th11: :: CLOPBAN4:11
for X being Complex_Banach_Algebra
for m being Nat
for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds
seq1 . n = seq2 . n ) holds
(Partial_Sums seq1) . m = (Partial_Sums seq2) . m
proof end;

theorem Th12: :: CLOPBAN4:12
for X being Complex_Banach_Algebra
for seq being sequence of X
for rseq being Real_Sequence st ( for n being Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 holds
( seq is convergent & lim seq = 0. X )
proof end;

definition
let X be Complex_Banach_Algebra;
let z be Element of X;
func z ExpSeq -> sequence of X means :Def1: :: CLOPBAN4:def 1
for n being Nat holds it . n = (1r / (n !)) * (z #N n);
existence
ex b1 being sequence of X st
for n being Nat holds b1 . n = (1r / (n !)) * (z #N n)
proof end;
uniqueness
for b1, b2 being sequence of X st ( for n being Nat holds b1 . n = (1r / (n !)) * (z #N n) ) & ( for n being Nat holds b2 . n = (1r / (n !)) * (z #N n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines ExpSeq CLOPBAN4:def 1 :
for X being Complex_Banach_Algebra
for z being Element of X
for b3 being sequence of X holds
( b3 = z ExpSeq iff for n being Nat holds b3 . n = (1r / (n !)) * (z #N n) );

scheme :: CLOPBAN4:sch 1
ExNormSpaceCASE{ F1() -> Complex_Banach_Algebra, F2( Nat, Nat) -> Point of F1() } :
for k being Nat ex seq being sequence of F1() st
for n being Nat holds
( ( n <= k implies seq . n = F2(k,n) ) & ( n > k implies seq . n = 0. F1() ) )
proof end;

definition
let n be Nat;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Expan (n,z,w) -> sequence of X means :Def2: :: CLOPBAN4:def 2
for k being Nat holds
( ( k <= n implies it . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );
existence
ex b1 being sequence of X st
for k being Nat holds
( ( k <= n implies b1 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b1 . k = 0. X ) )
proof end;
uniqueness
for b1, b2 being sequence of X st ( for k being Nat holds
( ( k <= n implies b1 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b1 . k = 0. X ) ) ) & ( for k being Nat holds
( ( k <= n implies b2 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines Expan CLOPBAN4:def 2 :
for n being Nat
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan (n,z,w) iff for k being Nat holds
( ( k <= n implies b5 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );

definition
let n be Nat;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Expan_e (n,z,w) -> sequence of X means :Def3: :: CLOPBAN4:def 3
for k being Nat holds
( ( k <= n implies it . k = ((() . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );
existence
ex b1 being sequence of X st
for k being Nat holds
( ( k <= n implies b1 . k = ((() . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b1 . k = 0. X ) )
proof end;
uniqueness
for b1, b2 being sequence of X st ( for k being Nat holds
( ( k <= n implies b1 . k = ((() . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b1 . k = 0. X ) ) ) & ( for k being Nat holds
( ( k <= n implies b2 . k = ((() . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines Expan_e CLOPBAN4:def 3 :
for n being Nat
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan_e (n,z,w) iff for k being Nat holds
( ( k <= n implies b5 . k = ((() . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );

definition
let n be Nat;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Alfa (n,z,w) -> sequence of X means :Def4: :: CLOPBAN4:def 4
for k being Nat holds
( ( k <= n implies it . k = (() . k) * (() . (n -' k)) ) & ( n < k implies it . k = 0. X ) );
existence
ex b1 being sequence of X st
for k being Nat holds
( ( k <= n implies b1 . k = (() . k) * (() . (n -' k)) ) & ( n < k implies b1 . k = 0. X ) )
proof end;
uniqueness
for b1, b2 being sequence of X st ( for k being Nat holds
( ( k <= n implies b1 . k = (() . k) * (() . (n -' k)) ) & ( n < k implies b1 . k = 0. X ) ) ) & ( for k being Nat holds
( ( k <= n implies b2 . k = (() . k) * (() . (n -' k)) ) & ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines Alfa CLOPBAN4:def 4 :
for n being Nat
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Alfa (n,z,w) iff for k being Nat holds
( ( k <= n implies b5 . k = (() . k) * (() . (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );

definition
let X be Complex_Banach_Algebra;
let z, w be Element of X;
let n be Nat;
func Conj (n,z,w) -> sequence of X means :Def5: :: CLOPBAN4:def 5
for k being Nat holds
( ( k <= n implies it . k = (() . k) * ((() . n) - (() . (n -' k))) ) & ( n < k implies it . k = 0. X ) );
existence
ex b1 being sequence of X st
for k being Nat holds
( ( k <= n implies b1 . k = (() . k) * ((() . n) - (() . (n -' k))) ) & ( n < k implies b1 . k = 0. X ) )
proof end;
uniqueness
for b1, b2 being sequence of X st ( for k being Nat holds
( ( k <= n implies b1 . k = (() . k) * ((() . n) - (() . (n -' k))) ) & ( n < k implies b1 . k = 0. X ) ) ) & ( for k being Nat holds
( ( k <= n implies b2 . k = (() . k) * ((() . n) - (() . (n -' k))) ) & ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines Conj CLOPBAN4:def 5 :
for X being Complex_Banach_Algebra
for z, w being Element of X
for n being Nat
for b5 being sequence of X holds
( b5 = Conj (n,z,w) iff for k being Nat holds
( ( k <= n implies b5 . k = (() . k) * ((() . n) - (() . (n -' k))) ) & ( n < k implies b5 . k = 0. X ) ) );

theorem Th13: :: CLOPBAN4:13
for X being Complex_Banach_Algebra
for z being Element of X
for n being Nat holds
( () . (n + 1) = ((1r / (n + 1)) * z) * (() . n) & () . 0 = 1. X & ||.(() . n).|| <= () . n )
proof end;

theorem :: CLOPBAN4:14
for X being Complex_Banach_Algebra
for k being Nat
for seq being sequence of X st 0 < k holds
(Shift seq) . k = seq . (k -' 1) by LOPBAN_4:15;

theorem Th15: :: CLOPBAN4:15
for X being Complex_Banach_Algebra
for k being Nat
for seq being sequence of X holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k)
proof end;

theorem Th16: :: CLOPBAN4:16
for X being Complex_Banach_Algebra
for n being Nat
for z, w being Element of X st z,w are_commutative holds
(z + w) #N n = (Partial_Sums (Expan (n,z,w))) . n
proof end;

theorem Th17: :: CLOPBAN4:17
for X being Complex_Banach_Algebra
for w, z being Element of X
for n being Nat holds Expan_e (n,z,w) = (1r / (n !)) * (Expan (n,z,w))
proof end;

theorem Th18: :: CLOPBAN4:18
for X being Complex_Banach_Algebra
for n being Nat
for z, w being Element of X st z,w are_commutative holds
(1r / (n !)) * ((z + w) #N n) = (Partial_Sums (Expan_e (n,z,w))) . n
proof end;

theorem Th19: :: CLOPBAN4:19
for X being Complex_Banach_Algebra holds
( (0. X) ExpSeq is norm_summable & Sum ((0. X) ExpSeq) = 1. X )
proof end;

registration
let X be Complex_Banach_Algebra;
let z be Element of X;
coherence
proof end;
end;

theorem Th20: :: CLOPBAN4:20
for X being Complex_Banach_Algebra
for w, z being Element of X holds
( () . 0 = 1. X & (Expan (0,z,w)) . 0 = 1. X )
proof end;

theorem Th21: :: CLOPBAN4:21
for X being Complex_Banach_Algebra
for w, z being Element of X
for k, l being Nat st l <= k holds
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
proof end;

theorem Th22: :: CLOPBAN4:22
for X being Complex_Banach_Algebra
for w, z being Element of X
for k being Nat holds (Partial_Sums (Alfa ((k + 1),z,w))) . k = ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k)
proof end;

theorem Th23: :: CLOPBAN4:23
for X being Complex_Banach_Algebra
for w, z being Element of X
for k being Nat holds () . k = (Expan_e (k,z,w)) . k
proof end;

theorem Th24: :: CLOPBAN4:24
for X being Complex_Banach_Algebra
for n being Nat
for z, w being Element of X st z,w are_commutative holds
(Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n
proof end;

theorem Th25: :: CLOPBAN4:25
for X being Complex_Banach_Algebra
for k being Nat
for z, w being Element of X st z,w are_commutative holds
((() . k) * (() . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k
proof end;

theorem Th26: :: CLOPBAN4:26
for X being Complex_Banach_Algebra
for z being Element of X
for n being Nat holds 0 <= () . n
proof end;

theorem Th27: :: CLOPBAN4:27
for X being Complex_Banach_Algebra
for z being Element of X
for k being Nat holds
( ||.(() . k).|| <= () . k & () . k <= Sum () & ||.(() . k).|| <= Sum () )
proof end;

theorem Th28: :: CLOPBAN4:28
for X being Complex_Banach_Algebra
for z being Element of X holds 1 <= Sum ()
proof end;

theorem Th29: :: CLOPBAN4:29
for X being Complex_Banach_Algebra
for z being Element of X
for m, n being Nat holds
( |.(() . n).| = () . n & ( n <= m implies |.((() . m) - (() . n)).| = (() . m) - (() . n) ) )
proof end;

theorem Th30: :: CLOPBAN4:30
for X being Complex_Banach_Algebra
for w, z being Element of X
for k, n being Nat holds |.((Partial_Sums ||.(Conj (k,z,w)).||) . n).| = (Partial_Sums ||.(Conj (k,z,w)).||) . n
proof end;

theorem Th31: :: CLOPBAN4:31
for X being Complex_Banach_Algebra
for w, z being Element of X
for p being Real st p > 0 holds
ex n being Nat st
for k being Nat st n <= k holds
|.((Partial_Sums ||.(Conj (k,z,w)).||) . k).| < p
proof end;

theorem Th32: :: CLOPBAN4:32
for X being Complex_Banach_Algebra
for w, z being Element of X
for seq being sequence of X st ( for k being Nat holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds
( seq is convergent & lim seq = 0. X )
proof end;

Lm3: for X being Complex_Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum ()) * (Sum ()) = Sum ((z + w) ExpSeq)

proof end;

definition
let X be Complex_Banach_Algebra;
func exp_ X -> Function of the carrier of X, the carrier of X means :Def6: :: CLOPBAN4:def 6
for z being Element of X holds it . z = Sum ();
existence
ex b1 being Function of the carrier of X, the carrier of X st
for z being Element of X holds b1 . z = Sum ()
proof end;
uniqueness
for b1, b2 being Function of the carrier of X, the carrier of X st ( for z being Element of X holds b1 . z = Sum () ) & ( for z being Element of X holds b2 . z = Sum () ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines exp_ CLOPBAN4:def 6 :
for X being Complex_Banach_Algebra
for b2 being Function of the carrier of X, the carrier of X holds
( b2 = exp_ X iff for z being Element of X holds b2 . z = Sum () );

definition
let X be Complex_Banach_Algebra;
let z be Element of X;
func exp z -> Element of X equals :: CLOPBAN4:def 7
(exp_ X) . z;
correctness
coherence
(exp_ X) . z is Element of X
;
;
end;

:: deftheorem defines exp CLOPBAN4:def 7 :
for X being Complex_Banach_Algebra
for z being Element of X holds exp z = (exp_ X) . z;

theorem :: CLOPBAN4:33
for X being Complex_Banach_Algebra
for z being Element of X holds exp z = Sum () by Def6;

theorem Th34: :: CLOPBAN4:34
for X being Complex_Banach_Algebra
for z1, z2 being Element of X st z1,z2 are_commutative holds
( exp (z1 + z2) = (exp z1) * (exp z2) & exp (z2 + z1) = (exp z2) * (exp z1) & exp (z1 + z2) = exp (z2 + z1) & exp z1, exp z2 are_commutative )
proof end;

theorem :: CLOPBAN4:35
for X being Complex_Banach_Algebra
for z1, z2 being Element of X st z1,z2 are_commutative holds
z1 * (exp z2) = (exp z2) * z1
proof end;

theorem Th36: :: CLOPBAN4:36
for X being Complex_Banach_Algebra holds exp (0. X) = 1. X
proof end;

theorem Th37: :: CLOPBAN4:37
for X being Complex_Banach_Algebra
for z being Element of X holds
( (exp z) * (exp (- z)) = 1. X & (exp (- z)) * (exp z) = 1. X )
proof end;

theorem :: CLOPBAN4:38
for X being Complex_Banach_Algebra
for z being Element of X holds
( exp z is invertible & (exp z) " = exp (- z) & exp (- z) is invertible & (exp (- z)) " = exp z )
proof end;

theorem Th39: :: CLOPBAN4:39
for X being Complex_Banach_Algebra
for z being Element of X
for s, t being Complex holds s * z,t * z are_commutative
proof end;

theorem :: CLOPBAN4:40
for X being Complex_Banach_Algebra
for z being Element of X
for s, t being Complex holds
( (exp (s * z)) * (exp (t * z)) = exp ((s + t) * z) & (exp (t * z)) * (exp (s * z)) = exp ((t + s) * z) & exp ((s + t) * z) = exp ((t + s) * z) & exp (s * z), exp (t * z) are_commutative )
proof end;