Now necessary condition for f(*) analytic is done by the Cauchy-Riemann relations...
What are the values of z [if any...] that satisfy (2)?...
I already know that it is real analytic in 1, with radius of convergence 1 and power series expansion:
I think that I can show that ln is real analytic for all points of (0,2). But I'd like to prove it for .
Thanks for your replies. But I cannot use any knowledge of complex numbers. It's only real analysis, no complex numbers.
I use the definition from Tao's Analysis II:
Let E be a subset of R and let f:E--> R be a function. If a is an interior point of E, we say that f is real analytic at a if there exists an open interval (a-r,a+r) in E for some r>0 such that there exists a power series centered at a which has a radius of convergence greater than or equal to r, and which converges to f on (a-r,a+r). If E is an open set, and f is real analytic at every point a of E, we say that f is real analytic on E.
(where open) is analytic in if and only if for all there exist an interval, and constants with the property that for all . The proof is not difficult, the key is using an appropiate form of the residue in Taylor's formula.
From this, and noting that the result follows.