05/31/2011, 11:00 PM
yeah , i was somewhat aware of all that.
more specifically a theorem by hadamard or was it weierstrass :
any entire function can be written as
f(z) = exp(taylor(z)) * (z - a_1)(z - a_2)..(z - a_n)...
i guess this implies if f(z)/f(0) is entire then
f(z)/f(0) = 1 + a_1 z + a_2 z^2 + ... = (1 + b_1 z)(1 + b_2 z^2) ...
and the product form also converges for all z apart from the set of points (-1/b_n)^(1/n) (unless they truely give 0 ) IFF those zero's are nowhere dense ( so that they do not form a " natural boundary " if there are oo many zero's ).
to avoid bounded non-cauchy sequences , i do assume a simple summability method is used ( averaging ).
( or "productability method " -> exp ( summability ( sum log (c_n) ) ) )
tommy1729
more specifically a theorem by hadamard or was it weierstrass :
any entire function can be written as
f(z) = exp(taylor(z)) * (z - a_1)(z - a_2)..(z - a_n)...
i guess this implies if f(z)/f(0) is entire then
f(z)/f(0) = 1 + a_1 z + a_2 z^2 + ... = (1 + b_1 z)(1 + b_2 z^2) ...
and the product form also converges for all z apart from the set of points (-1/b_n)^(1/n) (unless they truely give 0 ) IFF those zero's are nowhere dense ( so that they do not form a " natural boundary " if there are oo many zero's ).
to avoid bounded non-cauchy sequences , i do assume a simple summability method is used ( averaging ).
( or "productability method " -> exp ( summability ( sum log (c_n) ) ) )
tommy1729