05/31/2011, 12:26 PM
consider rewriting a taylor series f(z)/f(0) as a kind of taylor product :
f(z)/f(0) = 1 + a x + b x^2 + ... = (1 + a' x)(1 + b'x^2) ...
if we want (1 + a' x)(1 + b'x^2) ... to be valid at least where the taylor series is we have to " take care of the zero's "
(1 + c x^n)*... has as zero's (-1/c)^(1/n) so the taylor series must have these zero's too.
despite that simple condition , its not quite easy to me how to find
f(z)/f(0) = 1 + a x + b x^2 + ... = (1 + a' x)(1 + b'x^2) ...
with both expressions converging in the same domain ( or all of C ).
f(z)/f(0) = 1 + a x + b x^2 + ... = (1 + a' x)(1 + b'x^2) ...
if we want (1 + a' x)(1 + b'x^2) ... to be valid at least where the taylor series is we have to " take care of the zero's "
(1 + c x^n)*... has as zero's (-1/c)^(1/n) so the taylor series must have these zero's too.
despite that simple condition , its not quite easy to me how to find
f(z)/f(0) = 1 + a x + b x^2 + ... = (1 + a' x)(1 + b'x^2) ...
with both expressions converging in the same domain ( or all of C ).