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u ( φ , k ) = 0 φ d y 1 - k 2 sin 2 ( y )

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

s n ( u , k ) = sin ( φ ( u , k ) )

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

Figure two is a graph titled elliptic sine of u, modulus k. The horizontal axis is labeled Independent Variable, u, and ranges in variable from 0 to 12 in increments of 2. The vertical axis is labeled sn(u, k) and ranges in value from -1 to 1 in increments of 0.5. There are four wavelike functions on this graph. The first, a blue curve, begins increasing from the origin and completes two peaks and nearly two troughs before it terminates with a positive slope at (12, -0.5). This curve is labeled k=0. The second, a green curve, is wavelike and begins positive in a similar fashion as the blue curve, but its trough is substantially wider, and it only completes one and a half peaks and one full trough before it terminates at (12, 1) when it is just about to reach the top of a peak. This curve is labeled k=0.9. The third, a red curve, has an even wider peak than the previous curve, completing one peak and half of one trough before it terminates traveling completely horizontal in a portion of the trough at (12, -1). This curve is labeled k=0.99. The fourth, a teal curve, is again an exaggeration of a sinusoidal function with an extremely wide peak, that does not even reach the horizontal apex of the trough when it terminates at (12, -1). This curve is labeled k=0.999.
Jacobian Elliptic Sine Function of u with Modulus k

For k=0, s n ( u , 0 ) = sin ( u ) . As k approaches 1, the s n ( u , k ) looks like a "fat" sine function. For k = 1 , s n ( u , 1 ) = tanh ( u ) 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] .

Figure three is a graph titled complete elliptic integral, K_k. The horizontal axis is labeled modulus, k and ranges in value from 0 to 1 in increments of 1. The vertical axis is labeled complete elliptic integral, and ranges in value from 0 to 5 in increments of 0.5. There is one curve in this graph, and at its beginning at approximately (0, 1.5) is an arrow pointing at the place it begins, labeled π/2. The curve moves from left to right with a shallow slope at first, but the slope is slowly increasing across the page, and by the horizontal value 0.8 the graph has a sharp positive slope. At the horizontal value 1, the curve is completely vertical, and ends in the top-right corner of the graph, at (1, 5).
Complete Elliptic Integral as a function of the Modulus

For a modulus of zero, the quarter period is K = π / 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 ' defined by

k 2 + k ' 2 = 1

where both k and k ' are assumed real and between 0 and 1. The complete elliptic integral of the complementary modulus is denoted K ' .

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

c n ( u , k ) = c o s ( φ ( u , k ) )
s c ( u , k ) = t a n ( φ ( u , k ) )
c s ( u , k ) = c t n ( φ ( u , k ) )
n c ( u , k ) = s e c ( φ ( u , k ) )
n s ( u , k ) = c s c ( φ ( u , k ) )

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

d n ( u , k ) = 1 - k 2 s n 2 ( u , k )

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

s n 2 ( u , k ) + c n 2 ( u , k ) = 1

They have derivatives that are elliptic functions. For example,

d s n d u = c n d n

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

x ' ' + a x ± 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

s n ( j v , k ) = j s c ( v , k ' )
c n ( j v , k ) = n c ( v , k ' )

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 ' , respectively. They are the only class of functions that are “doubly periodic".

One particular value that the s n function takes on that is important in creating a rational function is

s n ( K + j K ' , k ) = 1 / k

The chebyshev rational function

The rational function G ( ω ) 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 ( ω ) = s n ( n s n - 1 ( ω , k ) , k 1 )

In terms of the intermediate complex variable φ , G ( ω ) and ω become

G ( ω ) = s n ( n φ , k 1 )
ω = s n ( φ , k )

It can be shown [link] that G ( ω ) 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 ( ω ) to the definition of the Chebyshev polynomial C N ( ω ) . In this case, however, n is not necessarily an integerand is not the order of the filter. Requiring that G ( ω ) 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:

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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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