Complex Numbers and Complex Algebra: Taylor Series
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The following theorem is the converse of the above theorem and says that every analytic function can be represented by a power series inside domain of analyticity.
Taylor Theorem: Let $f(z)$ be analytic in MATH. Then, $f(z)$ has a power series expansion around $z_{0}$ given by

MATH

where MATH for $n =0$, $1$, $2$, $\cdots$ where MATH for any $r$ with $0 < r < R$. This series is called the Taylor series of $f$ about the point $z_{0}$ and has radius of convergence $\ge R$. Further, the Taylor series of $f$ about that point $z_{0}$ is unique.

Note:
1. The Taylor series of $f(z)$ about the point $z_{0} = 0$ is called the Maclaurin series of $f$.
2. If $f(z)$ is analytic in MATH for some $R > 0$, then by Taylor theorem, $f(z)$ can be approximated with arbitrarily high precision by a polynomial $P_{n}(z)$ of sufficiently high degree.
   
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