# 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 $-1?$

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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