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

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- Nov 16th 2009, 01:26 AMbram kierkels[SOLVED] Logarithm real analytic
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 - Nov 16th 2009, 01:27 AMbram kierkels
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). - Nov 16th 2009, 06:29 AMchisigma
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

- Nov 16th 2009, 07:59 AMbram kierkels
- Nov 16th 2009, 08:21 AMchisigma