Taylor Series reaches closest Singularity

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Theorem

Let the singularities of a function be the points at which the function is not analytic.


Let $F$ be a complex function.

Let $F$ be analytic everywhere except at a finite number of singularities.

Let $x_0$ be a point in $\R$ where $F$ is analytic.

Let $R \in \R_{>0}$ be the distance from $x_0$ to the closest singularity of $F$.


Let $f = F {\restriction_\R}$ be a real function.


Then:

the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \in \R$ satisfying $\size {x - x_0} < R$


Proof

We have that $F$ is analytic everywhere except at its singularities.

Also, the distance from $x_0$ to the closest singularity of $F$ is $R$.

Therefore:

$F$ is analytic at every point $z \in \C$ satisfying $\size {z - x_0} < R$

The result follows by Convergence of Taylor Series of Function Analytic on Disk.

$\blacksquare$


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