# 1.3 Useful mathematical identities

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#### ${e}^{j\theta }$ , cos θ , and sin θ

${e}^{j\theta }=\underset{n\to \infty }{lim}{\left(1+j\frac{\theta }{n}\right)}^{n}=\sum n=0\infty \frac{1}{n!}{\left(j\theta \right)}^{n}=cos\theta +jsin\theta cos\theta =\sum n=0\infty \frac{{\left(-1\right)}^{n}}{\left(2n\right)!}{\theta }^{2n};sin\theta =\sum n=0\infty \frac{{\left(-1\right)}^{n}}{\left(2n+1\right)!}{\theta }^{2n+1}$

$cos\theta =\sum n=0\infty \frac{{\left(-1\right)}^{n}}{\left(2n\right)!}{\theta }^{2n};sin\theta =\sum n=0\infty \frac{{\left(-1\right)}^{n}}{\left(2n+1\right)!}{\theta }^{2n+1}$

#### Trigonometric identities

${sin}^{2}\theta +{cos}^{2}\theta =1$
$sin\left(\theta +\phi \right)=sin\theta cos\phi +cos\theta sin\phi$
$cos\left(\theta +\phi \right)=cos\theta cos\phi -sin\theta sin\phi$
$sin\left(\theta -\phi \right)=sin\theta cos\phi -cos\theta sin\phi$
$cos\left(\theta -\phi \right)=cos\theta cos\phi +sin\theta sin\phi$

#### Euler's equations

${e}^{j\theta }=cos\theta +jsin\theta$
$sin\theta =\frac{{e}^{j\theta }-{e}^{-j\theta }}{2j}$
$cos\theta =\frac{{e}^{j\theta }+{e}^{-j\theta }}{2}$

#### De moivre's identity

$\left(cos\theta +jsin{\theta \right)}^{n}=cosn\theta +jsinn\theta$

#### Binomial expansion

${\left(x+y\right)}^{N}=\sum n=0NNn{x}^{n}{y}^{N-n};Nn=\frac{N!}{\left(N-n\right)!n!}$
${2}^{N}=\sum n=0NNn$

#### Geometric sums

$\sum k=0\infty a{z}^{k}=\frac{a}{1-z}|z|<1$
$\sum k=0N-1a{z}^{k}=\frac{a\left(1-{z}^{N}\right)}{1-z}z\ne 1$

#### Taylor's series

$f\left(x\right)=\sum k=0\infty {f}^{\left(k\right)}\left(a\right)\frac{{\left(x-a\right)}^{k}}{k!}$
$\left(\text{Maclaurin's Series if}a=0\right)$

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why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Bharti
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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