when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?
when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?
A function is analytic if it has a convergent taylor series in a particular domain.
A convergent Taylor series which actually converges to the function! Take for example the classic . Then, is infinitely differentiable and its Maclaurin series congerges, but it doesn't converge to !
when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?
An important detail is that if a complex function is differentiable in , then it is analytic in and it is representable as Taylor function 'somewhere around' ... that isn't necessarly true for an 'ordinary' real function...