begin
definition
canceled;
end;
:: deftheorem CLOPBAN4:def 1 :
canceled;
Lm1:
for X being Complex_Banach_Algebra
for z being Element of X
for n being Element of 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 )
Lm2:
for X being Complex_Banach_Algebra
for n being Element of 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 )
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
:: deftheorem Def2 defines ExpSeq CLOPBAN4:def 2 :
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 Element of NAT holds b3 . n = (1r / (n !c )) * (z #N n) );
:: deftheorem CLOPBAN4:def 3 :
canceled;
:: deftheorem Def4 defines Expan CLOPBAN4:def 4 :
for n being Element of 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 Element of NAT holds
( ( k <= n implies b5 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def5 defines Expan_e CLOPBAN4:def 5 :
for n being Element of 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 Element of NAT holds
( ( k <= n implies b5 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def6 defines Alfa CLOPBAN4:def 6 :
for n being Element of 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 Element of NAT holds
( ( k <= n implies b5 . k = ((z ExpSeq ) . k) * ((Partial_Sums (w ExpSeq )) . (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def7 defines Conj CLOPBAN4:def 7 :
for X being Complex_Banach_Algebra
for z, w being Element of X
for n being Element of NAT
for b5 being sequence of X holds
( b5 = Conj n,z,w iff for k being Element of NAT holds
( ( k <= n implies b5 . k = ((z ExpSeq ) . k) * (((Partial_Sums (w ExpSeq )) . n) - ((Partial_Sums (w ExpSeq )) . (n -' k))) ) & ( n < k implies b5 . k = 0. X ) ) );
theorem Th13:
theorem
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
theorem Th32:
Lm3:
for X being Complex_Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum (z ExpSeq )) * (Sum (w ExpSeq )) = Sum ((z + w) ExpSeq )
:: deftheorem Def8 defines exp_ CLOPBAN4:def 8 :
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 (z ExpSeq ) );
:: deftheorem defines exp CLOPBAN4:def 9 :
for X being Complex_Banach_Algebra
for z being Element of X holds exp z = (exp_ X) . z;
theorem
theorem Th34:
theorem
theorem Th36:
theorem Th37:
theorem
theorem Th39:
theorem