I want to proof that the natural logarithm is real analytic.
For I have
Is it satisfied when I say when ,
I have
Because it is also real analytic when ?
Thanks
Definition Real Analytic;
F is real analytic at a if there excist an open interval (a-r, a+r) in E for some r>0 such that there exist a power series centered at a which has radius of convergence greater than or equal to r, and which converges to f on (a-r, a+r).
The 'natural logarithm' function is analytic in so that the Taylor expansion around this point exists and it is...
, (1)
If You want to 'expand' the range of definition of the function, there are two possibilities...
a) because is , writing instead to in the second term of (1) and changing sign You obtain...
, (2)
b) because is , writing instead of in the second term of (1) You obtain…
, (3)
Kind regards