# 3.4 Elliptic-function filter properties  (Page 2/5)

 Page 2 / 5
$u\left(\phi ,k\right)={\int }_{0}^{\phi }\frac{dy}{\sqrt{1-{k}^{2}{sin}^{2}\left(y\right)}}$

The trigonometric sine of the inverse of this function is defined as the Jacobian elliptic sine of $u$ with modulus $k$ , and is denoted

$sn\left(u,k\right)=sin\left(\phi \left(u,k\right)\right)$

A special evaluation of [link] is known as the complete elliptic integral $K=u\left(\pi /2,k\right)$ . It can be shown [link] that $sn\left(u\right)$ and most of the other elliptic functions are periodic with periods $4K$ if $u$ is real. Because of this, $K$ is also called the “quarter period". A plot of $sn\left(u,k\right)$ for several values of the modulus $k$ is shown in [link] .

For k=0, $sn\left(u,0\right)=sin\left(u\right)$ . As $k$ approaches 1, the $sn\left(u,k\right)$ looks like a "fat" sine function. For $k=1$ , $sn\left(u,1\right)=tanh\left(u\right)$ and is not periodic (period becomes infinite).

The quarter period or complete elliptic integral $K$ is a function of the modulus $k$ and is illustrated in [link] .

For a modulus of zero, the quarter period is $K=\pi /2$ and it does not increase much until k nears unity. It then increasesrapidly and goes to infinity as $k$ goes to unity.

Another parameter that is used is the complementary modulus ${k}^{\text{'}}$ defined by

${k}^{2}+{k}^{\text{'}2}=1$

where both $k$ and ${k}^{\text{'}}$ are assumed real and between 0 and 1. The complete elliptic integral of the complementary modulus is denoted ${K}^{\text{'}}$ .

In addition to the elliptic sine, other elliptic functions that are rather obvious generalizations are

$cn\left(u,k\right)=cos\left(\phi \left(u,k\right)\right)$
$sc\left(u,k\right)=tan\left(\phi \left(u,k\right)\right)$
$cs\left(u,k\right)=ctn\left(\phi \left(u,k\right)\right)$
$nc\left(u,k\right)=sec\left(\phi \left(u,k\right)\right)$
$ns\left(u,k\right)=csc\left(\phi \left(u,k\right)\right)$

There are six other elliptic functions that have no trigonometric counterparts [link] . One that is needed is

$dn\left(u,k\right)=\sqrt{1-{k}^{2}s{n}^{2}\left(u,k\right)}$

Many interesting properties of the elliptic functions exist [link] . They obey a large set of identities such as

$s{n}^{2}\left(u,k\right)+c{n}^{2}\left(u,k\right)=1$

They have derivatives that are elliptic functions. For example,

$\frac{d\phantom{\rule{4pt}{0ex}}sn}{du}=cn\phantom{\rule{4pt}{0ex}}dn$

The elliptic functions are the solutions of a set of nonlinear differential equations of the form

${x}^{\text{'}\text{'}}+ax±b{x}^{3}=0$

Some of the most important properties for the elliptic functions are as functions of a complex variable. For a purely imaginaryargument

$sn\left(jv,k\right)=jsc\left(v,{k}^{\text{'}}\right)$
$cn\left(jv,k\right)=nc\left(v,{k}^{\text{'}}\right)$

This indicates that the elliptic functions, in contrast to the circular and hyperbolic trigonometric functions, are periodic inboth the real and the imaginary part of the argument with periods related to $K$ and ${K}^{\text{'}}$ , respectively. They are the only class of functions that are “doubly periodic".

One particular value that the $sn$ function takes on that is important in creating a rational function is

$sn\left(K+j{K}^{\text{'}},k\right)=1/k$

## The chebyshev rational function

The rational function $G\left(\omega \right)$ needed in [link] is sometimes called a Chebyshev rational function because of its equal-ripple properties.It can be defined in terms of two elliptic functions with moduli $k$ and ${k}_{1}$ by

$G\left(\omega \right)=sn\phantom{\rule{4pt}{0ex}}\left(n\phantom{\rule{4pt}{0ex}}s{n}^{-1}\left(\omega ,k\right),{k}_{1}\right)$

In terms of the intermediate complex variable $\phi$ , $G\left(\omega \right)$ and $\omega$ become

$G\left(\omega \right)=sn\left(n\phi ,{k}_{1}\right)$
$\omega =sn\left(\phi ,k\right)$

It can be shown [link] that $G\left(\omega \right)$ is a real-valued rational function if the parameters $k$ , ${k}_{1}$ , and $n$ take on special values. Note the similarity of the definition of $G\left(\omega \right)$ to the definition of the Chebyshev polynomial ${C}_{N}\left(\omega \right)$ . In this case, however, n is not necessarily an integerand is not the order of the filter. Requiring that $G\left(\omega \right)$ be a rational function requires an alignment of the imaginary periods [link] of the two elliptic functions in [link] , [link] . It also requires alignment of an integer multiple of the real periods. The integermultiplier is denoted by $N$ and is the order of the resulting filter [link] . These two requirements are stated by the following very important relations:

Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
Got questions? Join the online conversation and get instant answers! By OpenStax By OpenStax By OpenStax By Stephen Voron By Janet Forrester By OpenStax By Katherina jennife... By Marion Cabalfin By JavaChamp Team By David Corey