I want to proof that the natural logarithm is real analytic.
For I have
Is it satisfied when I say when ,
Because it is also real analytic when ?
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...
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...
b) because is , writing instead of in the second term of (1) You obtain…