Let f be continu and real and satisfy :
For x = Y
\( f(1) = 1 \) (1)
\( f ' (x) = \Pi_1^Y f(x) \) (2)
\( f(2) = e \)
For real \( 1 < Y < 2 \)
1) does that imply f is C^oo in \( x E [1,Y] \) ?
2) does that imply f is analytic in \( [1,Y] \) ?
3) is the solution Unique ? If not, how many variables are there ?
4) how to prove existance ?
5) does it follow \( f ' (2x) = \Pi_1^{2Y} f(x) \) ?
6) is the equation \( f ' (x) = exp[ \Sigma 1^Y f(x-1) ] \) equivalent ?
7) does it Neccessarily follow \( f(x+1) = exp {f(x)} \) for x>1 ?
8 ) If f is analytic on \( [1,Y] \) does that imply it is analytic in \( [2,1+Y] \) ?
Regards
Tommy1729
For x = Y
\( f(1) = 1 \) (1)
\( f ' (x) = \Pi_1^Y f(x) \) (2)
\( f(2) = e \)
For real \( 1 < Y < 2 \)
1) does that imply f is C^oo in \( x E [1,Y] \) ?
2) does that imply f is analytic in \( [1,Y] \) ?
3) is the solution Unique ? If not, how many variables are there ?
4) how to prove existance ?
5) does it follow \( f ' (2x) = \Pi_1^{2Y} f(x) \) ?
6) is the equation \( f ' (x) = exp[ \Sigma 1^Y f(x-1) ] \) equivalent ?
7) does it Neccessarily follow \( f(x+1) = exp {f(x)} \) for x>1 ?
8 ) If f is analytic on \( [1,Y] \) does that imply it is analytic in \( [2,1+Y] \) ?
Regards
Tommy1729