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

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

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:

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Kristine 2*2*2=8
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J, combine like terms 7x-4y
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Samantha
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Asali
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
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Azam
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Prasenjit
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Damian
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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