# 3.8 Analytic functions and taylor series

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A summary of taylor series functions and their properties, including some practice exercises relating the taylor series to the identity theorem.

Let $S$ be a subset of $C,$ let $f:S\to C$ be a complex-valued function, and let $c$ be a point of $S.$ Then $f$ is said to be expandable in a Taylor series around c with radius of convergence $r$ if there exists an $r>0$ such that ${B}_{r}\left(c\right)\subseteq S,$ and $f\left(z\right)$ is given by the formula

$f\left(z\right)=\sum _{n=0}^{\infty }{a}_{n}{\left(z-c\right)}^{n}$

for all $z\in {B}_{r}\left(c\right).$

Let $S$ be a subset of $R,$ let $f:S\to R$ be a real-valued function on $S,$ and let $c$ be a point of $S.$ Then $f$ is said to be expandable in a Taylor series around c with radius of convergence $r$ if there exists an $r>0$ such that the interval $\left(c-r,c+r\right)\subseteq S,$ and $f\left(x\right)$ is given by the formula

$f\left(x\right)=\sum _{n=0}^{\infty }{a}_{n}{\left(x-c\right)}^{n}$

for all $x\in \left(c-r,c+r\right).$

Suppose $S$ is an open subset of $C.$ A function $f:S\to C$ is called analytic on S if it is expandable in a Taylor series around every point $c$ of $S.$

Suppose $S$ is an open subset of $R.$ A function $f:S\to C$ is called real analytic on S if it is expandable in a Taylor series around every point $c$ of $S.$

Suppose $S$ is a subset of $C,$ that $f:S\to C$ is a complex-valued function and that $c$ belongs to $S.$ Assume that $f$ is expandable in a Taylor series around $c$ with radius of convergence $r.$ Then $f$ is continuous at each $z\in {B}_{r}\left(c\right).$

Suppose $S$ is a subset of $R,$ that $f:S\to R$ is a real-valued function and that $c$ belongs to $S.$ Assume that $f$ is expandable in a Taylor series around $c$ with radius of convergence $r.$ Then $f$ is continuous at each $x\in \left(c-r,c+r\right).$

If we let $g$ be the power series function given by $g\left(z\right)=\sum {a}_{n}{z}^{n},$ and $T$ be the function defined by $T\left(z\right)=z-c,$ then $f\left(z\right)=g\left(T\left(z\right)\right),$ and this theorem is a consequence of [link] and [link] .

Prove that $f\left(z\right)=1/z$ is analytic on its domain.

HINT: Use $r=|c|,$ and then use the infinite geometric series.

State and prove an Identity Theorem, analogous to [link] , for functions that are expandable in a Taylor series around a point $c.$

1. Prove that every polynomial is expandable in a Taylor series around every point $c.$ HINT: Use the binomial theorem.
2. Is the exponential function expandable in a Taylor series around the number $-1?$

#### Questions & Answers

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Almas
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da
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Professor
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Professor
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LITNING Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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