First, you probably noticed that there is a typo: the equality is in fact , as in claim 2.

Then, I agree that the justification is not well-written, as several points in this note, I have to update it.

A possibility is to notice from the proof of the analyticity of holomorphic functions that, if a function is holomorphic on the disk , then it can be written as a power series on for every . Now, let ; we know that is holomorphic on , and on the other hand, we know that on . But if you write as a power series on , with , because it agrees with the previous one on , they are in fact identical. Therefore, the series converges for all , where ; equivalently, the series converges for all .

]]>In fact, I don’t see why will have an expansion centred at which includes the points .

I agree that there is an expansion in the smaller disc though.

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