Complex Numbers and Complex Algebra: Taylor Series
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3. If $f(z)$ is analytic at $z_{0}$, then the radius of convergence $R$ of the Taylor series of $f(z)$ about $z = z_{0}$ is the distance from $z_{0}$ to the point (singularity) nearest to $z_{0}$ at which $f(z)$ ceases to be analytic.

Important Note:
1. By Taylor theorem, if a function $f(z)$ is infinitely differentiable in an open set $D$, then $f(z)$ can be expanded in power series in $D$. This result is not true in case of real valued functions of a real variable. For example, the function MATH for MATH and $f(0) = 0$ is infinitely many times differentiable in the neighborhood of $x_{0} = 0$, but $f(x)$ can not be represented by a power series about the point $x_{0} = 0$.
2. A real Taylor series of a real valued function $f$ of a real variable converges if and only if the Taylor remainder term goes to zero. In a complex Taylor series, the remainder term is irrelevant; the Taylor series will converge to in the largest disk that one can fit inside the domain of analyticity of $f$.

Example: Find the power series of $f(z) = e^{z}$ about the point $z_{0} = i$.
Observe that the $n$-th derivative $f^{(n)}(z) = e^{z}$ for $z \in \QTR{Bbb}{C}$. This gives that $f^{(n)}(i) = e^{i}$ for MATH. Therefore, the Taylor series of MATH and it has radius of convergence $R = \infty$.

   
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