Setting is...
(1)
Now necessary condition for f(*) analytic is done by the Cauchy-Riemann relations...
(2)
What are the values of z [if any...] that satisfy (2)?...
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I already know that it is real analytic in 1, with radius of convergence 1 and power series expansion:
for
Now what?
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 .
Some thoughts?
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.
That's not true... the logarithm has an essential singularity at the origin, of the worst possible kind on top of that! The largest possible domain on which a fixed branch of a logarithm can live comfortably is the plane cut along some curve joining and . But you're right in the sense that the existence of a holomorphic function on which coincides with the usual (real) logarithm on the real line immediately implies that the usual logarithm is analytic there (and everywhere else on ).
That is not fully exact!... the function is a multivalued function that in has a singularity classified as branch point. Effectively such type of singularity is, in a certain sense, 'the worst possible' because for an with an essential singularity in [like for example ...] it is possible to obtain the Laurent expansion around , but for that is impossible...
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For more information about the singularity of in see...
Logarithmic Singularity -- from Wolfram MathWorld
Kind regards