In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular,[1] but his work was published much later.
There are twelve Jacobi elliptic functions denoted by , where and are any of the letters , , , and . (Functions of the form are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both and .[2] The distribution of the zeros and poles in the -plane is well-known. However, questions of the distribution of the zeros and poles in the -plane remain to be investigated.[2]
In the complex plane of the argument , the twelve functions form a repeating lattice of simple poles and zeroes.[3] Depending on the function, one repeating parallelogram, or unit cell, will have sides of length or on the real axis, and or on the imaginary axis, where and are known as the quarter periods with being the elliptic integral of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin at one corner, and as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named , , , and , going counter-clockwise from the origin. The function will have a zero at the corner and a pole at the corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.
When the argument and parameter are real, with , and will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.
Since the Jacobian elliptic functions are doubly periodic in , they factor through a torus – in effect, their domain can be taken to be a torus, just as cosine and sine are in effect defined on a circle. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary. The complex plane can be replaced by a complex torus. The circumference of the first circle is and the second , where and are the quarter periods. Each function has two zeroes and two poles at opposite positions on the torus. Among the points , , , there is one zero and one pole.
The Jacobian elliptic functions are then doubly periodic, meromorphic functions satisfying the following properties:
There is a simple zero at the corner , and a simple pole at the corner .
The complex number is equal to half the period of the function ; that is, the function is periodic in the direction , with the period being . The function is also periodic in the other two directions and , with periods such that and are quarter periods.
Jacobi elliptic function
Jacobi elliptic function
Jacobi elliptic function
Jacobi elliptic function
Plots of four Jacobi Elliptic Functions in the complex plane of , illustrating their double periodic behavior. Images generated using a version of the domain coloring method.[4] All have values of equal to .
Notationedit
The elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing. Elliptic functions are functions of two variables. The first variable might be given in terms of the amplitude, or more commonly, in terms of given below. The second variable might be given in terms of the parameter, or as the elliptic modulus, where , or in terms of the modular angle, where . The complements of and are defined as and . These four terms are used below without comment to simplify various expressions.
The twelve Jacobi elliptic functions are generally written as where and are any of the letters , , , and . Functions of the form are trivially set to unity for notational completeness. The “major” functions are generally taken to be , and from which all other functions can be derived and expressions are often written solely in terms of these three functions, however, various symmetries and generalizations are often most conveniently expressed using the full set. (This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.)
Throughout this article, .
The functions are notationally related to each other by the multiplication rule: (arguments suppressed)
from which other commonly used relationships can be derived:
The multiplication rule follows immediately from the identification of the elliptic functions with the Neville theta functions[5]
Also note that:
Definition in terms of inverses of elliptic integralsedit
There is a definition, relating the elliptic functions to the inverse of the incomplete elliptic integral of the first kind. These functions take the parameters and as inputs. The that satisfies
is called the Jacobi amplitude:
In this framework, the elliptic sine sn u (Latin: sinus amplitudinis) is given by
and the elliptic cosine cn u (Latin: cosinus amplitudinis) is given by
and the delta amplitude dn u (Latin: delta amplitudinis)[note 1]
In the above, the value is a free parameter, usually taken to be real such that , and so the elliptic functions can be thought of as being given by two variables, and the parameter . The remaining nine elliptic functions are easily built from the above three (, , ), and are given in a section below.
In the most general setting, is a multivalued function (in ) with infinitely many logarithmic branch points (the branches differ by integer multiples of ), namely the points and where .[6] This multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making analytic everywhere except on the branch cuts. In contrast, and other elliptic functions have no branch points, give consistent values for every branch of , and are meromorphic in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), (when considered as a single-valued function) is not an elliptic function.
However, the integral inversion above defines a unique single-valued real-analytic function in a real neighborhood of if is real. There is a unique analytic continuation of this function from that neighborhood to . The analytic continuation of this function is periodic in if and only if (with the minimal period ), and it is denoted by in the rest of this article.
The Jacobi epsilon function is not an elliptic function. However, unlike the Jacobi amplitude and coamplitude, the Jacobi epsilon function is meromorphic in the whole complex plane (in both and ).
The Jacobi zn function is defined by
It is a singly periodic function which is meromorphic in . Its minimal period is . It is related to the Jacobi zeta function by
Definition as trigonometry: the Jacobi ellipseedit
are defined on the unit circle, with radius r = 1 and angle arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse,[citation needed] with a = 1. Let
then:
For each angle the parameter
(the incomplete elliptic integral of the first kind) is computed. On the unit circle (), would be an arc length. The quantity is related to the incomplete elliptic integral of the second kind (with modulus ) by[8]
and therefore is related to the arc length of an ellipse. Let be a point on the ellipse, and let be the point where the unit circle intersects the line between and the origin . Then the familiar relations from the unit circle:
read for the ellipse:
So the projections of the intersection point of the line with the unit circle on the x- and y-axes are simply and . These projections may be interpreted as 'definition as trigonometry'. In short:
For the and value of the point with and parameter we get, after inserting the relation:
into: that:
The latter relations for the x- and y-coordinates of points on the unit ellipse may be considered as generalization of the relations for the coordinates of points on the unit circle.
The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ,dn) with
Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn}
q
c
s
n
d
p
c
1
s
1
n
1
d
1
Definition in terms of Jacobi theta functionsedit
Jacobi theta function descriptionedit
Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate as , and respectively as (the theta constants) then the theta function elliptic modulusk is . We define the nome as in relation to the period ratio. We have
The Jacobi zn function can be expressed by theta functions as well:
where denotes the derivative with respect to the first variable.
Elliptic integral and elliptic nomeedit
Since the Jacobi functions are defined in terms of the elliptic modulus , we need to invert this and find in terms of . We start from , the complementary modulus. As a function of it is
These are two identical definitions of the complete elliptic integral of the first kind:
An identical definition of the nome function can me produced by using a series. Following function has this identity:
Since we may reduce to the case where the imaginary part of is greater than or equal to (see Modular group), we can assume the absolute value of is less than or equal to ; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for . By solving this function after q we get:[10][11][12]
Simplifications of complicated products of the Jacobi elliptic functions are often made easier using these identities.
Jacobi transformationsedit
The Jacobi imaginary transformationsedit
The Jacobi imaginary transformations relate various functions of the imaginary variable i u or, equivalently, relations between various values of the m parameter. In terms of the major functions:[14]: 506
Using the multiplication rule, all other functions may be expressed in terms of the above three. The transformations may be generally written as . The following table gives the for the specified pq(u,m).[13] (The arguments are suppressed)
Jacobi Imaginary transformations
q
c
s
n
d
p
c
1
i ns
nc
nd
s
−i sn
1
−i sc
−i sd
n
cn
i cs
1
cd
d
dn
i ds
dc
1
Since the hyperbolic trigonometric functions are proportional to the circular trigonometric functions with imaginary arguments, it follows that the Jacobi functions will yield the hyperbolic functions for m=1.[5]: 249 In the figure, the Jacobi curve has degenerated to two vertical lines at x = 1 and x = −1.
The Jacobi real transformationsedit
The Jacobi real transformations[5]: 308 yield expressions for the elliptic functions in terms with alternate values of m. The transformations may be generally written as . The following table gives the for the specified pq(u,m).[13] (The arguments are suppressed)
Jacobi real transformations
q
c
s
n
d
p
c
s
n
d
Other Jacobi transformationsedit
Jacobi's real and imaginary transformations can be combined in various ways to yield three more simple transformations .[5]: 214 The real and imaginary transformations are two transformations in a group (D3 or anharmonic group) of six transformations. If
is the transformation for the m parameter in the real transformation, and
is the transformation of m in the imaginary transformation, then the other transformations can be built up by successive application of these two basic transformations, yielding only three more possibilities:
These five transformations, along with the identity transformation (μU(m) = m) yield the six-element group. With regard to the Jacobi elliptic functions, the general transformation can be expressed using just three functions:
where i = U, I, IR, R, RI, or RIR, identifying the transformation, γi is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other nine transformed functions can be built up from the above three. The reason the cs, ns, ds functions were chosen to represent the transformation is that the other functions will be ratios of these three (except for their inverses) and the multiplication factors will cancel.
The following table lists the multiplication factors for the three ps functions, the transformed m's, and the transformed function names for each of the six transformations.[5]: 214 (As usual, k2 = m, 1 − k2 = k12 = m′ and the arguments () are suppressed)
Parameters for the six transformations
Transformation i
cs'
ns'
ds'
U
1
m
cs
ns
ds
I
i
m'
ns
cs
ds
IR
i k
−m'/m
ds
cs
ns
R
k
1/m
ds
ns
cs
RI
i k1
1/m'
ns
ds
cs
RIR
k1
−m/m'
cs
ds
ns
Thus, for example, we may build the following table for the RIR transformation.[13] The transformation is generally written (The arguments are suppressed)
The RIR transformation
q
c
s
n
d
p
c
1
k' cs
cd
cn
s
sc
1
sd
sn
n
dc
ds
1
dn
d
nc
ns
nd
1
The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any complex-valued parameter m can be converted into another set for which 0 ≤ m ≤ 1 and, for real values of u, the function values will be real.[5]: p. 215
The Jacobi hyperbolaedit
Introducing complex numbers, our ellipse has an associated hyperbola:
from applying Jacobi's imaginary transformation[13] to the elliptic functions in the above equation for x and y.
It follows that we can put . So our ellipse has a dual ellipse with m replaced by 1-m. This leads to the complex torus mentioned in the Introduction.[15] Generally, m may be a complex number, but when m is real and m<0, the curve is an ellipse with major axis in the x direction. At m=0 the curve is a circle, and for 0<m<1, the curve is an ellipse with major axis in the y direction. At m = 1, the curve degenerates into two vertical lines at x = ±1. For m > 1, the curve is a hyperbola. When m is complex but not real, x or y or both are complex and the curve cannot be described on a real x-y diagram.
Minor functionsedit
Reversing the order of the two letters of the function name results in the reciprocals of the three functions above:
Similarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:
More compactly, we have
where p and q are any of the letters s, c, d.
Periodicity, poles, and residuesedit
In the complex plane of the argument u, the Jacobi elliptic functions form a repeating pattern of poles (and zeroes). The residues of the poles all have the same absolute value, differing only in sign. Each function pq(u,m) has an "inverse function" (in the multiplicative sense) qp(u,m) in which the positions of the poles and zeroes are exchanged. The periods of repetition are generally different in the real and imaginary directions, hence the use of the term "doubly periodic" to describe them.
The Jacobi amplitude and the Jacobi epsilon function are quasi-periodic:
The double periodicity of the Jacobi elliptic functions may be expressed as:
where α and β are any pair of integers. K(·) is the complete elliptic integral of the first kind, also known as the quarter period. The power of negative unity (γ) is given in the following table:
q
c
s
n
d
p
c
0
β
α + β
α
s
β
0
α
α + β
n
α + β
α
0
β
d
α
α + β
β
0
When the factor (−1)γ is equal to −1, the equation expresses quasi-periodicity. When it is equal to unity, it expresses full periodicity. It can be seen, for example, that for the entries containing only α when α is even, full periodicity is expressed by the above equation, and the function has full periods of 4K(m) and 2iK(1 − m). Likewise, functions with entries containing only β have full periods of 2K(m) and 4iK(1 − m), while those with α + β have full periods of 4K(m) and 4iK(1 − m).
In the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged. The pole and zero arrangement in the auxiliary rectangle formed by (0,0), (K,0), (0,K′) and (K,K′) are in accordance with the description of the pole and zero placement described in the introduction above. Also, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole. The residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:
Residues of Jacobi Elliptic Functions
q
c
s
n
d
p
c
1
s
n
1
d
-1
1
When applicable, poles displaced above by 2K or displaced to the right by 2K′ have the same value but with signs reversed, while those diagonally opposite have the same value. Note that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.
To get x3, we take the tangent of twice the arctangent of the modulus.
Also this equation leads to the sn-value of the third of K:
These equations lead to the other values of the Jacobi-Functions:
Fifth K formula
Following equation has following solution:
To get the sn-values, we put the solution x into following expressions:
Relations between squares of the functionsedit
Relations between squares of the functions can be derived from two basic relationships (Arguments (u,m) suppressed):
where m + m' = 1. Multiplying by any function of the form nq yields more general equations:
With q = d, these correspond trigonometrically to the equations for the unit circle () and the unit ellipse (), with x = cd, y = sd and r = nd. Using the multiplication rule, other relationships may be derived. For example:
Addition theoremsedit
The functions satisfy the two square relations (dependence on m supressed)
From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions[3]
April 09, 2024
jacobi, elliptic, functions, mathematics, basic, elliptic, functions, they, found, description, motion, pendulum, also, pendulum, mathematics, well, design, electronic, elliptic, filters, while, trigonometric, functions, defined, with, reference, circle, gener. In mathematics the Jacobi elliptic functions are a set of basic elliptic functions They are found in the description of the motion of a pendulum see also pendulum mathematics as well as in the design of electronic elliptic filters While trigonometric functions are defined with reference to a circle the Jacobi elliptic functions are a generalization which refer to other conic sections the ellipse in particular The relation to trigonometric functions is contained in the notation for example by the matching notation sn displaystyle operatorname sn for sin displaystyle sin The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and or understood They were introduced by Carl Gustav Jakob Jacobi 1829 Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797 the lemniscate elliptic functions in particular 1 but his work was published much later Contents 1 Overview 2 Notation 3 Definition in terms of inverses of elliptic integrals 4 Definition as trigonometry the Jacobi ellipse 5 Definition in terms of Jacobi theta functions 5 1 Jacobi theta function description 5 2 Elliptic integral and elliptic nome 6 Definition in terms of Neville theta functions 7 Jacobi transformations 7 1 The Jacobi imaginary transformations 7 2 The Jacobi real transformations 7 3 Other Jacobi transformations 8 The Jacobi hyperbola 9 Minor functions 10 Periodicity poles and residues 11 Special values 12 Identities 12 1 Half Angle formula 12 2 K formulas 12 3 Relations between squares of the functions 12 4 Addition theorems 13 Jacobi elliptic functions as solutions of nonlinear ordinary differential equations 14 Expansion in terms of the nome 15 Fast computation 16 Approximation in terms of hyperbolic functions 17 Continued fractions 18 Inverse functions 19 Map projection 20 See also 21 Notes 22 References 23 External linksOverview edit nbsp The fundamental rectangle in the complex plane of u displaystyle u nbsp There are twelve Jacobi elliptic functions denoted by pq u m displaystyle operatorname pq u m nbsp where p displaystyle mathrm p nbsp and q displaystyle mathrm q nbsp are any of the letters c displaystyle mathrm c nbsp s displaystyle mathrm s nbsp n displaystyle mathrm n nbsp and d displaystyle mathrm d nbsp Functions of the form pp u m displaystyle operatorname pp u m nbsp are trivially set to unity for notational completeness u displaystyle u nbsp is the argument and m displaystyle m nbsp is the parameter both of which may be complex In fact the Jacobi elliptic functions are meromorphic in both u displaystyle u nbsp and m displaystyle m nbsp 2 The distribution of the zeros and poles in the u displaystyle u nbsp plane is well known However questions of the distribution of the zeros and poles in the m displaystyle m nbsp plane remain to be investigated 2 In the complex plane of the argument u displaystyle u nbsp the twelve functions form a repeating lattice of simple poles and zeroes 3 Depending on the function one repeating parallelogram or unit cell will have sides of length 2K displaystyle 2K nbsp or 4K displaystyle 4K nbsp on the real axis and 2K displaystyle 2K nbsp or 4K displaystyle 4K nbsp on the imaginary axis where K K m displaystyle K K m nbsp and K K 1 m displaystyle K K 1 m nbsp are known as the quarter periods with K displaystyle K cdot nbsp being the elliptic integral of the first kind The nature of the unit cell can be determined by inspecting the auxiliary rectangle generally a parallelogram which is a rectangle formed by the origin 0 0 displaystyle 0 0 nbsp at one corner and K K displaystyle K K nbsp as the diagonally opposite corner As in the diagram the four corners of the auxiliary rectangle are named s displaystyle mathrm s nbsp c displaystyle mathrm c nbsp d displaystyle mathrm d nbsp and n displaystyle mathrm n nbsp going counter clockwise from the origin The function pq u m displaystyle operatorname pq u m nbsp will have a zero at the p displaystyle mathrm p nbsp corner and a pole at the q displaystyle mathrm q nbsp corner The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle When the argument u displaystyle u nbsp and parameter m displaystyle m nbsp are real with 0 lt m lt 1 displaystyle 0 lt m lt 1 nbsp K displaystyle K nbsp and K displaystyle K nbsp will be real and the auxiliary parallelogram will in fact be a rectangle and the Jacobi elliptic functions will all be real valued on the real line Since the Jacobian elliptic functions are doubly periodic in u displaystyle u nbsp they factor through a torus in effect their domain can be taken to be a torus just as cosine and sine are in effect defined on a circle Instead of having only one circle we now have the product of two circles one real and the other imaginary The complex plane can be replaced by a complex torus The circumference of the first circle is 4K displaystyle 4K nbsp and the second 4K displaystyle 4K nbsp where K displaystyle K nbsp and K displaystyle K nbsp are the quarter periods Each function has two zeroes and two poles at opposite positions on the torus Among the points 0 displaystyle 0 nbsp K displaystyle K nbsp K iK displaystyle K iK nbsp iK displaystyle iK nbsp there is one zero and one pole The Jacobian elliptic functions are then doubly periodic meromorphic functions satisfying the following properties There is a simple zero at the corner p displaystyle mathrm p nbsp and a simple pole at the corner q displaystyle mathrm q nbsp The complex number p q displaystyle mathrm p mathrm q nbsp is equal to half the period of the function pq u displaystyle operatorname pq u nbsp that is the function pq u displaystyle operatorname pq u nbsp is periodic in the direction pq displaystyle operatorname pq nbsp with the period being 2 p q displaystyle 2 mathrm p mathrm q nbsp The function pq u displaystyle operatorname pq u nbsp is also periodic in the other two directions pp displaystyle mathrm pp nbsp and pq displaystyle mathrm pq nbsp with periods such that p p displaystyle mathrm p mathrm p nbsp and p q displaystyle mathrm p mathrm q nbsp are quarter periods nbsp Jacobi elliptic function sn displaystyle operatorname sn nbsp nbsp Jacobi elliptic function cn displaystyle operatorname cn nbsp nbsp Jacobi elliptic function dn displaystyle operatorname dn nbsp nbsp Jacobi elliptic function sc displaystyle operatorname sc nbsp Plots of four Jacobi Elliptic Functions in the complex plane of u displaystyle u nbsp illustrating their double periodic behavior Images generated using a version of the domain coloring method 4 All have values of k m displaystyle k sqrt m nbsp equal to 0 8 displaystyle 0 8 nbsp Notation editThe elliptic functions can be given in a variety of notations which can make the subject unnecessarily confusing Elliptic functions are functions of two variables The first variable might be given in terms of the amplitude f displaystyle varphi nbsp or more commonly in terms of u displaystyle u nbsp given below The second variable might be given in terms of the parameter m displaystyle m nbsp or as the elliptic modulus k displaystyle k nbsp where k2 m displaystyle k 2 m nbsp or in terms of the modular angle a displaystyle alpha nbsp where m sin2 a displaystyle m sin 2 alpha nbsp The complements of k displaystyle k nbsp and m displaystyle m nbsp are defined as m 1 m displaystyle m 1 m nbsp and k m textstyle k sqrt m nbsp These four terms are used below without comment to simplify various expressions The twelve Jacobi elliptic functions are generally written as pq u m displaystyle operatorname pq u m nbsp where p displaystyle mathrm p nbsp and q displaystyle mathrm q nbsp are any of the letters c displaystyle mathrm c nbsp s displaystyle mathrm s nbsp n displaystyle mathrm n nbsp and d displaystyle mathrm d nbsp Functions of the form pp u m displaystyle operatorname pp u m nbsp are trivially set to unity for notational completeness The major functions are generally taken to be cn u m displaystyle operatorname cn u m nbsp sn u m displaystyle operatorname sn u m nbsp and dn u m displaystyle operatorname dn u m nbsp from which all other functions can be derived and expressions are often written solely in terms of these three functions however various symmetries and generalizations are often most conveniently expressed using the full set This notation is due to Gudermann and Glaisher and is not Jacobi s original notation Throughout this article pq u t2 pq u t displaystyle operatorname pq u t 2 operatorname pq u t nbsp The functions are notationally related to each other by the multiplication rule arguments suppressed pq p q pq p q displaystyle operatorname pq cdot operatorname p q operatorname pq cdot operatorname p q nbsp from which other commonly used relationships can be derived prqr pq displaystyle frac operatorname pr operatorname qr operatorname pq nbsp pr rq pq displaystyle operatorname pr cdot operatorname rq operatorname pq nbsp 1qp pq displaystyle frac 1 operatorname qp operatorname pq nbsp The multiplication rule follows immediately from the identification of the elliptic functions with the Neville theta functions 5 pq u m 8p u m 8q u m displaystyle operatorname pq u m frac theta operatorname p u m theta operatorname q u m nbsp Also note that K m K k2 01dt 1 t2 1 mt2 01dt 1 t2 1 k2t2 displaystyle K m K k 2 int 0 1 frac dt sqrt 1 t 2 1 mt 2 int 0 1 frac dt sqrt 1 t 2 1 k 2 t 2 nbsp Definition in terms of inverses of elliptic integrals edit nbsp Model of the Jacobi amplitude measured along vertical axis as a function of independent variables u and the modulus kThere is a definition relating the elliptic functions to the inverse of the incomplete elliptic integral of the first kind These functions take the parameters u displaystyle u nbsp and m displaystyle m nbsp as inputs The f displaystyle varphi nbsp that satisfies u 0fd81 msin2 8 displaystyle u int 0 varphi frac mathrm d theta sqrt 1 m sin 2 theta nbsp is called the Jacobi amplitude am u m f displaystyle operatorname am u m varphi nbsp In this framework the elliptic sine sn u Latin sinus amplitudinis is given by sn u m sin am u m displaystyle operatorname sn u m sin operatorname am u m nbsp and the elliptic cosine cn u Latin cosinus amplitudinis is given by cn u m cos am u m displaystyle operatorname cn u m cos operatorname am u m nbsp and the delta amplitude dn u Latin delta amplitudinis note 1 dn u m dduam u m displaystyle operatorname dn u m frac mathrm d mathrm d u operatorname am u m nbsp In the above the value m displaystyle m nbsp is a free parameter usually taken to be real such that 0 m 1 displaystyle 0 leq m leq 1 nbsp and so the elliptic functions can be thought of as being given by two variables u displaystyle u nbsp and the parameter m displaystyle m nbsp The remaining nine elliptic functions are easily built from the above three sn displaystyle operatorname sn nbsp cn displaystyle operatorname cn nbsp dn displaystyle operatorname dn nbsp and are given in a section below In the most general setting am u m displaystyle operatorname am u m nbsp is a multivalued function in u displaystyle u nbsp with infinitely many logarithmic branch points the branches differ by integer multiples of 2p displaystyle 2 pi nbsp namely the points 2sK m 4t 1 K 1 m i displaystyle 2sK m 4t 1 K 1 m i nbsp and 2sK m 4t 3 K 1 m i displaystyle 2sK m 4t 3 K 1 m i nbsp where s t Z displaystyle s t in mathbb Z nbsp 6 This multivalued function can be made single valued by cutting the complex plane along the line segments joining these branch points the cutting can be done in non equivalent ways giving non equivalent single valued functions thus making am u m displaystyle operatorname am u m nbsp analytic everywhere except on the branch cuts In contrast sin am u m displaystyle sin operatorname am u m nbsp and other elliptic functions have no branch points give consistent values for every branch of am displaystyle operatorname am nbsp and are meromorphic in the whole complex plane Since every elliptic function is meromorphic in the whole complex plane by definition am u m displaystyle operatorname am u m nbsp when considered as a single valued function is not an elliptic function However the integral inversion above defines a unique single valued real analytic function in a real neighborhood of u 0 displaystyle u 0 nbsp if m displaystyle m nbsp is real There is a unique analytic continuation of this function from that neighborhood to u R displaystyle u in mathbb R nbsp The analytic continuation of this function is periodic in u displaystyle u nbsp if and only if m gt 1 displaystyle m gt 1 nbsp with the minimal period 4K 1 m m displaystyle 4K 1 m sqrt m nbsp and it is denoted by am u m displaystyle operatorname am u m nbsp in the rest of this article Jacobi also introduced the coamplitude function coam u m am K m u m displaystyle operatorname coam u m operatorname am K m u m nbsp The Jacobi epsilon function can be defined as 7 E u m 0udn2 t m dt displaystyle mathcal E u m int 0 u operatorname dn 2 t m mathrm d t nbsp and relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind with parameter m displaystyle m nbsp E f m E F f m m displaystyle E varphi m mathcal E F varphi m m nbsp The Jacobi epsilon function is not an elliptic function However unlike the Jacobi amplitude and coamplitude the Jacobi epsilon function is meromorphic in the whole complex plane in both u displaystyle u nbsp and m displaystyle m nbsp The Jacobi zn function is defined by zn u m 0u dn t m 2 E m K m dt displaystyle operatorname zn u m int 0 u left operatorname dn t m 2 frac E m K m right mathrm d t nbsp It is a singly periodic function which is meromorphic in u displaystyle u nbsp Its minimal period is 2K m displaystyle 2K m nbsp It is related to the Jacobi zeta function by Z f m zn F f m m displaystyle Z varphi m operatorname zn F varphi m m nbsp Note that when f p 2 displaystyle varphi pi 2 nbsp that u displaystyle u nbsp then equals the quarter period K displaystyle K nbsp Definition as trigonometry the Jacobi ellipse edit nbsp Plot of the Jacobi ellipse x2 y2 b2 1 b real and the twelve Jacobi elliptic functions pq u m for particular values of angle f and parameter b The solid curve is the ellipse with m 1 1 b2 and u F f m where F is the elliptic integral of the first kind with parameter m k2 displaystyle m k 2 nbsp The dotted curve is the unit circle Tangent lines from the circle and ellipse at x cd crossing the x axis at dc are shown in light grey cos f sin f displaystyle cos varphi sin varphi nbsp are defined on the unit circle with radius r 1 and angle f displaystyle varphi nbsp arc length of the unit circle measured from the positive x axis Similarly Jacobi elliptic functions are defined on the unit ellipse citation needed with a 1 Let x2 y2b2 1 b gt 1 m 1 1b2 0 lt m lt 1 x rcos f y rsin f displaystyle begin aligned amp x 2 frac y 2 b 2 1 quad b gt 1 amp m 1 frac 1 b 2 quad 0 lt m lt 1 amp x r cos varphi quad y r sin varphi end aligned nbsp then r f m 11 msin2 f displaystyle r varphi m frac 1 sqrt 1 m sin 2 varphi nbsp For each angle f displaystyle varphi nbsp the parameter u u f m 0fr 8 m d8 displaystyle u u varphi m int 0 varphi r theta m d theta nbsp the incomplete elliptic integral of the first kind is computed On the unit circle a b 1 displaystyle a b 1 nbsp u displaystyle u nbsp would be an arc length The quantity u f k u f k2 displaystyle u varphi k u varphi k 2 nbsp is related to the incomplete elliptic integral of the second kind with modulus k displaystyle k nbsp by 8 u f k 11 k2 1 1 k22E f arctan 1 k2tan f 1 1 k21 1 k2 E f k k2sin fcos f21 k2sin2 f displaystyle u varphi k frac 1 sqrt 1 k 2 left frac 1 sqrt 1 k 2 2 operatorname E left varphi arctan left sqrt 1 k 2 tan varphi right frac 1 sqrt 1 k 2 1 sqrt 1 k 2 right operatorname E varphi k frac k 2 sin varphi cos varphi 2 sqrt 1 k 2 sin 2 varphi right nbsp and therefore is related to the arc length of an ellipse Let P x y rcos f rsin f displaystyle P x y r cos varphi r sin varphi nbsp be a point on the ellipse and let P x y cos f sin f displaystyle P x y cos varphi sin varphi nbsp be the point where the unit circle intersects the line between P displaystyle P nbsp and the origin O displaystyle O nbsp Then the familiar relations from the unit circle x cos f y sin f displaystyle x cos varphi quad y sin varphi nbsp read for the ellipse x cn u m y sn u m displaystyle x operatorname cn u m quad y operatorname sn u m nbsp So the projections of the intersection point P displaystyle P nbsp of the line OP displaystyle OP nbsp with the unit circle on the x and y axes are simply cn u m displaystyle operatorname cn u m nbsp and sn u m displaystyle operatorname sn u m nbsp These projections may be interpreted as definition as trigonometry In short cn u m xr f m sn u m yr f m dn u m 1r f m displaystyle operatorname cn u m frac x r varphi m quad operatorname sn u m frac y r varphi m quad operatorname dn u m frac 1 r varphi m nbsp For the x displaystyle x nbsp and y displaystyle y nbsp value of the point P displaystyle P nbsp with u displaystyle u nbsp and parameter m displaystyle m nbsp we get after inserting the relation r f m 1dn u m displaystyle r varphi m frac 1 operatorname dn u m nbsp into x r f m cos f y r f m sin f displaystyle x r varphi m cos varphi y r varphi m sin varphi nbsp that x cn u m dn u m y sn u m dn u m displaystyle x frac operatorname cn u m operatorname dn u m quad y frac operatorname sn u m operatorname dn u m nbsp The latter relations for the x and y coordinates of points on the unit ellipse may be considered as generalization of the relations x cos f y sin f displaystyle x cos varphi y sin varphi nbsp for the coordinates of points on the unit circle The following table summarizes the expressions for all Jacobi elliptic functions pq u m in the variables x y r and f dn with r x2 y2 textstyle r sqrt x 2 y 2 nbsp Jacobi elliptic functions pq u m as functions of x y r and f dn qc s n dpc 1 x y cot f displaystyle x y cot varphi nbsp x r cos f displaystyle x r cos varphi nbsp x cos f dn displaystyle x cos varphi operatorname dn nbsp s y x tan f displaystyle y x tan varphi nbsp 1 y r sin f displaystyle y r sin varphi nbsp y sin f dn displaystyle y sin varphi operatorname dn nbsp n r x sec f displaystyle r x sec varphi nbsp r y csc f displaystyle r y csc varphi nbsp 1 r 1 dn displaystyle r 1 operatorname dn nbsp d 1 x sec f dn displaystyle 1 x sec varphi operatorname dn nbsp 1 y csc f dn displaystyle 1 y csc varphi operatorname dn nbsp 1 r dn displaystyle 1 r operatorname dn nbsp 1Definition in terms of Jacobi theta functions editJacobi theta function description edit Equivalently Jacobi s elliptic functions can be defined in terms of his theta functions If we abbreviate ϑ00 0 q displaystyle vartheta 00 0 q nbsp as ϑ00 q displaystyle vartheta 00 q nbsp and ϑ01 0 q ϑ10 0 q ϑ11 0 q displaystyle vartheta 01 0 q vartheta 10 0 q vartheta 11 0 q nbsp respectively as ϑ01 q ϑ10 q ϑ11 q displaystyle vartheta 01 q vartheta 10 q vartheta 11 q nbsp the theta constants then the theta function elliptic modulus k is k ϑ10 q k ϑ00 q k 2 displaystyle k biggl vartheta 10 q k over vartheta 00 q k biggr 2 nbsp We define the nome as q exp pit displaystyle q exp pi i tau nbsp in relation to the period ratio We have sn u k ϑ00 q ϑ11 z q ϑ10 q ϑ01 z q cn u k ϑ01 q ϑ10 z q ϑ10 q ϑ01 z q dn u k ϑ01 q ϑ00 z q ϑ00 q ϑ01 z q displaystyle begin aligned operatorname sn u k amp frac vartheta 00 q vartheta 11 zeta q vartheta 10 q vartheta 01 zeta q 7pt operatorname cn u k amp frac vartheta 01 q vartheta 10 zeta q vartheta 10 q vartheta 01 zeta q 7pt operatorname dn u k amp frac vartheta 01 q vartheta 00 zeta q vartheta 00 q vartheta 01 zeta q end aligned nbsp where z pu 2K displaystyle zeta pi u 2K nbsp Edmund Whittaker and George Watson defined the Jacobi theta functions this way in their textbook A Course of Modern Analysis 9 ϑ00 v w n 1 1 w2n 1 2cos 2v w2n 1 w4n 2 displaystyle vartheta 00 v w prod n 1 infty 1 w 2n 1 2 cos 2v w 2n 1 w 4n 2 nbsp ϑ01 v w n 1 1 w2n 1 2cos 2v w2n 1 w4n 2 displaystyle vartheta 01 v w prod n 1 infty 1 w 2n 1 2 cos 2v w 2n 1 w 4n 2 nbsp ϑ10 v w 2w1 4cos v n 1 1 w2n 1 2cos 2v w2n w4n displaystyle vartheta 10 v w 2w 1 4 cos v prod n 1 infty 1 w 2n 1 2 cos 2v w 2n w 4n nbsp ϑ11 v w 2w1 4sin v n 1 1 w2n 1 2cos 2v w2n w4n displaystyle vartheta 11 v w 2w 1 4 sin v prod n 1 infty 1 w 2n 1 2 cos 2v w 2n w 4n nbsp The Jacobi zn function can be expressed by theta functions as well zn u k p2Kϑ01 z q ϑ01 z q p2Kϑ00 z q ϑ00 z q k2sn u k cn u k dn u k p2Kϑ10 z q ϑ10 z q dn u k sn u k cn u k p2Kϑ11 z q ϑ11 z q cn u k dn u k sn u k displaystyle begin aligned operatorname zn u k amp frac pi 2K frac vartheta 01 zeta q vartheta 01 zeta q amp frac pi 2K frac vartheta 00 zeta q vartheta 00 zeta q k 2 frac operatorname sn u k operatorname cn u k operatorname dn u k amp frac pi 2K frac vartheta 10 zeta q vartheta 10 zeta q frac operatorname dn u k operatorname sn u k operatorname cn u k amp frac pi 2K frac vartheta 11 zeta q vartheta 11 zeta q frac operatorname cn u k operatorname dn u k operatorname sn u k end aligned nbsp where displaystyle nbsp denotes the derivative with respect to the first variable Elliptic integral and elliptic nome edit Since the Jacobi functions are defined in terms of the elliptic modulus k t displaystyle k tau nbsp we need to invert this and find t displaystyle tau nbsp in terms of k displaystyle k nbsp We start from k 1 k2 displaystyle k sqrt 1 k 2 nbsp the complementary modulus As a function of t displaystyle tau nbsp it is k t 1 k2 ϑ01 q k ϑ00 q k 2 displaystyle k tau sqrt 1 k 2 biggl vartheta 01 q k over vartheta 00 q k biggr 2 nbsp Let us define the elliptic nome and the complete elliptic integral of the first kind q k exp pK 1 k2 K k displaystyle q k exp biggl pi frac K sqrt 1 k 2 K k biggr nbsp These are two identical definitions of the complete elliptic integral of the first kind K k 0p 211 k2sin f 2 f displaystyle K k int 0 pi 2 frac 1 sqrt 1 k 2 sin varphi 2 partial varphi nbsp K k p2 a 0 2a 216a a 4k2a displaystyle K k frac pi 2 sum a 0 infty frac 2a 2 16 a a 4 k 2a nbsp An identical definition of the nome function can me produced by using a series Following function has this identity 1 1 k241 1 k24 ϑ00 q k ϑ01 q k ϑ00 q k ϑ01 q k n 1 2q k 2n 1 2 1 n 1 2q k 4n2 1 displaystyle frac 1 sqrt 4 1 k 2 1 sqrt 4 1 k 2 frac vartheta 00 q k vartheta 01 q k vartheta 00 q k vartheta 01 q k biggl sum n 1 infty 2 q k 2n 1 2 biggr biggl 1 sum n 1 infty 2 q k 4n 2 biggr 1 nbsp Since we may reduce to the case where the imaginary part of t displaystyle tau nbsp is greater than or equal to 3 2 displaystyle sqrt 3 2 nbsp see Modular group we can assume the absolute value of q displaystyle q nbsp is less than or equal to exp p3 2 0 0658 displaystyle exp pi sqrt 3 2 approx 0 0658 nbsp for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q displaystyle q nbsp By solving this function after q we get 10 11 12 q k n 1 Sw n 24n 3 1 1 k241 1 k24 4n 3 k2 12 n 1 Sw n 1 24n 1k2n 4 displaystyle q k sum n 1 infty frac text Sw n 2 4n 3 biggl frac 1 sqrt 4 1 k 2 1 sqrt 4 1 k 2 biggr 4n 3 k 2 biggl frac 1 2 biggl sum n 1 infty frac text Sw n 1 2 4n 1 k 2n biggr biggr 4 nbsp Where SW n is sequence A002103 in the OEIS Definition in terms of Neville theta functions editThe Jacobi elliptic functions can be defined very simply using the Neville theta functions 13 pq u m 8p u m 8q u m displaystyle operatorname pq u m frac theta operatorname p u m theta operatorname q u m nbsp Simplifications of complicated products of the Jacobi elliptic functions are often made easier using these identities Jacobi transformations editThe Jacobi imaginary transformations edit nbsp Plot of the degenerate Jacobi curve x2 y2 b2 1 b and the twelve Jacobi Elliptic functions pq u 1 for a particular value of angle f The solid curve is the degenerate ellipse x2 1 with m 1 and u F f 1 where F is the elliptic integral of the first kind The dotted curve is the unit circle Since these are the Jacobi functions for m 0 circular trigonometric functions but with imaginary arguments they correspond to the six hyperbolic trigonometric functions The Jacobi imaginary transformations relate various functions of the imaginary variable i u or equivalently relations between various values of the m parameter In terms of the major functions 14 506 cn u m nc iu 1 m displaystyle operatorname cn u m operatorname nc i u 1 m nbsp sn u m isc iu 1 m displaystyle operatorname sn u m i operatorname sc i u 1 m nbsp dn u m dc iu 1 m displaystyle operatorname dn u m operatorname dc i u 1 m nbsp Using the multiplication rule all other functions may be expressed in terms of the above three The transformations may be generally written as pq u m gpqpq iu 1 m displaystyle operatorname pq u m gamma operatorname pq operatorname pq i u 1 m nbsp The following table gives the gpqpq iu 1 m displaystyle gamma operatorname pq operatorname pq i u 1 m nbsp for the specified pq u m 13 The arguments iu 1 m displaystyle i u 1 m nbsp are suppressed Jacobi Imaginary transformations gpqpq iu 1 m displaystyle gamma operatorname pq operatorname pq i u 1 m nbsp qc s n dpc 1 i ns nc nds i sn 1 i sc i sdn cn i cs 1 cdd dn i ds dc 1Since the hyperbolic trigonometric functions are proportional to the circular trigonometric functions with imaginary arguments it follows that the Jacobi functions will yield the hyperbolic functions for m 1 5 249 In the figure the Jacobi curve has degenerated to two vertical lines at x 1 and x 1 The Jacobi real transformations edit The Jacobi real transformations 5 308 yield expressions for the elliptic functions in terms with alternate values of m The transformations may be generally written as pq u m gpqpq ku 1 m displaystyle operatorname pq u m gamma operatorname pq operatorname pq k u 1 m nbsp The following table gives the gpqpq ku 1 m displaystyle gamma operatorname pq operatorname pq k u 1 m nbsp for the specified pq u m 13 The arguments ku 1 m displaystyle k u 1 m nbsp are suppressed Jacobi real transformations gpqpq ku 1 m displaystyle gamma operatorname pq operatorname pq k u 1 m nbsp qc s n dpc 1 displaystyle 1 nbsp kds displaystyle k operatorname ds nbsp dn displaystyle operatorname dn nbsp dc displaystyle operatorname dc nbsp s 1ksd displaystyle frac 1 k operatorname sd nbsp 1 displaystyle 1 nbsp 1ksn displaystyle frac 1 k operatorname sn nbsp 1ksc displaystyle frac 1 k operatorname sc nbsp n nd displaystyle operatorname nd nbsp kns displaystyle k operatorname ns nbsp 1 displaystyle 1 nbsp nc displaystyle operatorname nc nbsp d cd displaystyle operatorname cd nbsp kcs displaystyle k operatorname cs nbsp cn displaystyle operatorname cn nbsp 1 displaystyle 1 nbsp Other Jacobi transformations edit Jacobi s real and imaginary transformations can be combined in various ways to yield three more simple transformations 5 214 The real and imaginary transformations are two transformations in a group D3 or anharmonic group of six transformations If mR m 1 m displaystyle mu R m 1 m nbsp is the transformation for the m parameter in the real transformation and mI m 1 m m displaystyle mu I m 1 m m nbsp is the transformation of m in the imaginary transformation then the other transformations can be built up by successive application of these two basic transformations yielding only three more possibilities mIR m mI mR m m mmRI m mR mI m 1 m mRIR m mR mI mR m m m displaystyle begin aligned mu IR m amp amp mu I mu R m amp amp m m mu RI m amp amp mu R mu I m amp amp 1 m mu RIR m amp amp mu R mu I mu R m amp amp m m end aligned nbsp These five transformations along with the identity transformation mU m m yield the six element group With regard to the Jacobi elliptic functions the general transformation can be expressed using just three functions cs u m gics giu mi m displaystyle operatorname cs u m gamma i operatorname cs gamma i u mu i m nbsp ns u m gins giu mi m displaystyle operatorname ns u m gamma i operatorname ns gamma i u mu i m nbsp ds u m gids giu mi m displaystyle operatorname ds u m gamma i operatorname ds gamma i u mu i m nbsp where i U I IR R RI or RIR identifying the transformation gi is a multiplication factor common to these three functions and the prime indicates the transformed function The other nine transformed functions can be built up from the above three The reason the cs ns ds functions were chosen to represent the transformation is that the other functions will be ratios of these three except for their inverses and the multiplication factors will cancel The following table lists the multiplication factors for the three ps functions the transformed m s and the transformed function names for each of the six transformations 5 214 As usual k2 m 1 k2 k12 m and the arguments giu mi m displaystyle gamma i u mu i m nbsp are suppressed Parameters for the six transformations Transformation i gi displaystyle gamma i nbsp mi m displaystyle mu i m nbsp cs ns ds U 1 m cs ns dsI i m ns cs dsIR i k m m ds cs nsR k 1 m ds ns csRI i k1 1 m ns ds csRIR k1 m m cs ds nsThus for example we may build the following table for the RIR transformation 13 The transformation is generally written pq u m gpqpq k u m m displaystyle operatorname pq u m gamma operatorname pq operatorname pq k u m m nbsp The arguments k u m m displaystyle k u m m nbsp are suppressed The RIR transformation gpqpq k u m m displaystyle gamma operatorname pq operatorname pq k u m m nbsp qc s n dpc 1 k cs cd cns 1k displaystyle frac 1 k nbsp sc 1 1k displaystyle frac 1 k nbsp sd 1k displaystyle frac 1 k nbsp snn dc k displaystyle k nbsp ds 1 dnd nc k displaystyle k nbsp ns nd 1The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any complex valued parameter m can be converted into another set for which 0 m 1 and for real values of u the function values will be real 5 p 215 The Jacobi hyperbola edit nbsp Plot of the Jacobi hyperbola x2 y2 b2 1 b imaginary and the twelve Jacobi Elliptic functions pq u m for particular values of angle f and parameter b The solid curve is the hyperbola with m 1 1 b2 and u F f m where F is the elliptic integral of the first kind The dotted curve is the unit circle For the ds dc triangle s sin f cos f Introducing complex numbers our ellipse has an associated hyperbola x2 y2b2 1 displaystyle x 2 frac y 2 b 2 1 nbsp from applying Jacobi s imaginary transformation 13 to the elliptic functions in the above equation for x and y x 1dn u 1 m y sn u 1 m dn u 1 m displaystyle x frac 1 operatorname dn u 1 m quad y frac operatorname sn u 1 m operatorname dn u 1 m nbsp It follows that we can put x dn u 1 m y sn u 1 m displaystyle x operatorname dn u 1 m y operatorname sn u 1 m nbsp So our ellipse has a dual ellipse with m replaced by 1 m This leads to the complex torus mentioned in the Introduction 15 Generally m may be a complex number but when m is real and m lt 0 the curve is an ellipse with major axis in the x direction At m 0 the curve is a circle and for 0 lt m lt 1 the curve is an ellipse with major axis in the y direction At m 1 the curve degenerates into two vertical lines at x 1 For m gt 1 the curve is a hyperbola When m is complex but not real x or y or both are complex and the curve cannot be described on a real x y diagram Minor functions editReversing the order of the two letters of the function name results in the reciprocals of the three functions above ns u 1sn u nc u 1cn u nd u 1dn u displaystyle operatorname ns u frac 1 operatorname sn u qquad operatorname nc u frac 1 operatorname cn u qquad operatorname nd u frac 1 operatorname dn u nbsp Similarly the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator sc u sn u cn u sd u sn u dn u dc u dn u cn u ds u dn u sn u cs u cn u sn u cd u cn u dn u displaystyle begin aligned operatorname sc u frac operatorname sn u operatorname cn u qquad operatorname sd u frac operatorname sn u operatorname dn u qquad operatorname dc u frac operatorname dn u operatorname cn u qquad operatorname ds u frac operatorname dn u operatorname sn u qquad operatorname cs u frac operatorname cn u operatorname sn u qquad operatorname cd u frac operatorname cn u operatorname dn u end aligned nbsp More compactly we have pq u pn u qn u displaystyle operatorname pq u frac operatorname pn u operatorname qn u nbsp where p and q are any of the letters s c d Periodicity poles and residues edit nbsp Plots of the phase for the twelve Jacobi Elliptic functions pq u m as a function complex argument u with poles and zeroes indicated The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right which replaces the trivial dd function Regions with absolute value below 1 3 are colored black roughly indicating the location of a zero while regions with absolute value above 3 are colored white roughly indicating the position of a pole All plots use m 2 3 with K K m K K 1 m K being the complete elliptic integral of the first kind Arrows at the poles point in direction of zero phase Right and left arrows imply positive and negative real residues respectively Up and down arrows imply positive and negative imaginary residues respectively In the complex plane of the argument u the Jacobi elliptic functions form a repeating pattern of poles and zeroes The residues of the poles all have the same absolute value differing only in sign Each function pq u m has an inverse function in the multiplicative sense qp u m in which the positions of the poles and zeroes are exchanged The periods of repetition are generally different in the real and imaginary directions hence the use of the term doubly periodic to describe them The Jacobi amplitude and the Jacobi epsilon function are quasi periodic am u 2K m am u m p displaystyle operatorname am u 2K m operatorname am u m pi nbsp E u 2K m E u m 2E displaystyle mathcal E u 2K m mathcal E u m 2E nbsp where E m displaystyle E m nbsp is the complete elliptic integral of the second kind with parameter m displaystyle m nbsp Also zn u 2iK m zn u m piK displaystyle operatorname zn u 2iK m operatorname zn u m frac pi i K nbsp The double periodicity of the Jacobi elliptic functions may be expressed as pq u 2aK m 2ibK 1 m m 1 gpq u m displaystyle operatorname pq u 2 alpha K m 2i beta K 1 m m 1 gamma operatorname pq u m nbsp where a and b are any pair of integers K is the complete elliptic integral of the first kind also known as the quarter period The power of negative unity g is given in the following table g displaystyle gamma nbsp qc s n dpc 0 b a b as b 0 a a bn a b a 0 bd a a b b 0When the factor 1 g is equal to 1 the equation expresses quasi periodicity When it is equal to unity it expresses full periodicity It can be seen for example that for the entries containing only a when a is even full periodicity is expressed by the above equation and the function has full periods of 4K m and 2iK 1 m Likewise functions with entries containing only b have full periods of 2K m and 4iK 1 m while those with a b have full periods of 4K m and 4iK 1 m In the diagram on the right which plots one repeating unit for each function indicating phase along with the location of poles and zeroes a number of regularities can be noted The inverse of each function is opposite the diagonal and has the same size unit cell with poles and zeroes exchanged The pole and zero arrangement in the auxiliary rectangle formed by 0 0 K 0 0 K and K K are in accordance with the description of the pole and zero placement described in the introduction above Also the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole The residues of the poles closest to the origin in the figure i e in the auxiliary rectangle are listed in the following table Residues of Jacobi Elliptic Functions qc s n dpc 1 ik displaystyle frac i k nbsp 1k displaystyle frac 1 k nbsp s 1k displaystyle frac 1 k nbsp 1k displaystyle frac 1 k nbsp ikk displaystyle frac i k k nbsp n 1k displaystyle frac 1 k nbsp 1 ik displaystyle frac i k nbsp d 1 1 i displaystyle i nbsp When applicable poles displaced above by 2K or displaced to the right by 2K have the same value but with signs reversed while those diagonally opposite have the same value Note that poles and zeroes on the left and lower edges are considered part of the unit cell while those on the upper and right edges are not Special values editSetting m 1 displaystyle m 1 nbsp gives the lemniscate elliptic functions sl displaystyle operatorname sl nbsp and cl displaystyle operatorname cl nbsp sl u sn u 1 cl u cd u 1 cn u 1 dn u 1 displaystyle operatorname sl u operatorname sn u 1 quad operatorname cl u operatorname cd u 1 frac operatorname cn u 1 operatorname dn u 1 nbsp When m 0 displaystyle m 0 nbsp or m 1 displaystyle m 1 nbsp the Jacobi elliptic functions are reduced to non elliptic functions Function m 0 m 1sn u m displaystyle operatorname sn u m nbsp sin u displaystyle sin u nbsp tanh u displaystyle tanh u nbsp cn u m displaystyle operatorname cn u m nbsp cos u displaystyle cos u nbsp sech u displaystyle operatorname sech u nbsp dn u m displaystyle operatorname dn u m nbsp 1 displaystyle 1 nbsp sech u displaystyle operatorname sech u nbsp ns u m displaystyle operatorname ns u m nbsp csc u displaystyle csc u nbsp coth u displaystyle coth u nbsp nc u m displaystyle operatorname nc u m nbsp sec u displaystyle sec u nbsp cosh u displaystyle cosh u nbsp nd u m displaystyle operatorname nd u m nbsp 1 displaystyle 1 nbsp cosh u displaystyle cosh u nbsp sd u m displaystyle operatorname sd u m nbsp sin u displaystyle sin u nbsp sinh u displaystyle sinh u nbsp cd u m displaystyle operatorname cd u m nbsp cos u displaystyle cos u nbsp 1 displaystyle 1 nbsp cs u m displaystyle operatorname cs u m nbsp cot u displaystyle cot u nbsp csch u displaystyle operatorname csch u nbsp ds u m displaystyle operatorname ds u m nbsp csc u displaystyle csc u nbsp csch u displaystyle operatorname csch u nbsp dc u m displaystyle operatorname dc u m nbsp sec u displaystyle sec u nbsp 1 displaystyle 1 nbsp sc u m displaystyle operatorname sc u m nbsp tan u displaystyle tan u nbsp sinh u displaystyle sinh u nbsp For the Jacobi amplitude am u 0 u displaystyle operatorname am u 0 u nbsp and am u 1 gd u displaystyle operatorname am u 1 operatorname gd u nbsp where gd displaystyle operatorname gd nbsp is the Gudermannian function In general if neither of p q is d then pq u 1 pq gd u 0 displaystyle operatorname pq u 1 operatorname pq operatorname gd u 0 nbsp Identities editHalf Angle formula edit sn u2 m 1 cn u m 1 dn u m displaystyle operatorname sn left frac u 2 m right pm sqrt frac 1 operatorname cn u m 1 operatorname dn u m nbsp cn u2 m cn u m dn u m 1 dn u m displaystyle operatorname cn left frac u 2 m right pm sqrt frac operatorname cn u m operatorname dn u m 1 operatorname dn u m nbsp cn u2 m m dn u m mcn u m 1 dn u m displaystyle operatorname cn left frac u 2 m right pm sqrt frac m operatorname dn u m m operatorname cn u m 1 operatorname dn u m nbsp K formulas edit Half K formulasn 12K k k 21 k 1 k displaystyle operatorname sn left tfrac 1 2 K k k right frac sqrt 2 sqrt 1 k sqrt 1 k nbsp cn 12K k k 21 k241 k 1 k displaystyle operatorname cn left tfrac 1 2 K k k right frac sqrt 2 sqrt 4 1 k 2 sqrt 1 k sqrt 1 k nbsp dn 12K k k 1 k24 displaystyle operatorname dn left tfrac 1 2 K k k right sqrt 4 1 k 2 nbsp Third K formula sn 13K x3x6 1 1 x3x6 1 1 2x4 x2 1 x2 2 x2 1 12x4 x2 1 x2 2 x2 1 1 displaystyle operatorname sn left frac 1 3 K left frac x 3 sqrt x 6 1 1 right frac x 3 sqrt x 6 1 1 right frac sqrt 2 sqrt x 4 x 2 1 x 2 2 sqrt x 2 1 1 sqrt 2 sqrt x 4 x 2 1 x 2 2 sqrt x 2 1 1 nbsp To get x3 we take the tangent of twice the arctangent of the modulus Also this equation leads to the sn value of the third of K k2s4 2k2s3 2s 1 0 displaystyle k 2 s 4 2k 2 s 3 2s 1 0 nbsp s sn 13K k k displaystyle s operatorname sn left tfrac 1 3 K k k right nbsp These equations lead to the other values of the Jacobi Functions cn 23K k k 1 sn 13K k k displaystyle operatorname cn left tfrac 2 3 K k k right 1 operatorname sn left tfrac 1 3 K k k right nbsp dn 23K k k 1 sn 13K k k 1 displaystyle operatorname dn left tfrac 2 3 K k k right 1 operatorname sn left tfrac 1 3 K k k right 1 nbsp Fifth K formulaFollowing equation has following solution 4k2x6 8k2x5 2 1 k2 2x 1 k2 2 0 displaystyle 4k 2 x 6 8k 2 x 5 2 1 k 2 2 x 1 k 2 2 0 nbsp x 12 12k2sn 25K k k 2sn 45K k k 2 sn 45K k k 2 sn 25K k k 22sn 25K k k sn 45K k k displaystyle x frac 1 2 frac 1 2 k 2 operatorname sn left tfrac 2 5 K k k right 2 operatorname sn left tfrac 4 5 K k k right 2 frac operatorname sn left frac 4 5 K k k right 2 operatorname sn left frac 2 5 K k k right 2 2 operatorname sn left frac 2 5 K k k right operatorname sn left frac 4 5 K k k right nbsp To get the sn values we put the solution x into following expressions sn 25K k k 1 k2 1 22 1 x x2 x2 1 xx2 1 displaystyle operatorname sn left tfrac 2 5 K k k right 1 k 2 1 2 sqrt 2 1 x x 2 x 2 1 x sqrt x 2 1 nbsp sn 45K k k 1 k2 1 22 1 x x2 x2 1 xx2 1 displaystyle operatorname sn left tfrac 4 5 K k k right 1 k 2 1 2 sqrt 2 1 x x 2 x 2 1 x sqrt x 2 1 nbsp Relations between squares of the functions edit Relations between squares of the functions can be derived from two basic relationships Arguments u m suppressed cn2 sn2 1 displaystyle operatorname cn 2 operatorname sn 2 1 nbsp cn2 m sn2 dn2 displaystyle operatorname cn 2 m operatorname sn 2 operatorname dn 2 nbsp where m m 1 Multiplying by any function of the form nq yields more general equations cq2 sq2 nq2 displaystyle operatorname cq 2 operatorname sq 2 operatorname nq 2 nbsp cq2 m sq2 dq2 displaystyle operatorname cq 2 m operatorname sq 2 operatorname dq 2 nbsp With q d these correspond trigonometrically to the equations for the unit circle x2 y2 r2 displaystyle x 2 y 2 r 2 nbsp and the unit ellipse x2 m y2 1 displaystyle x 2 m y 2 1 nbsp with x cd y sd and r nd Using the multiplication rule other relationships may be derived For example dn2 m mcn2 msn2 m displaystyle operatorname dn 2 m m operatorname cn 2 m operatorname sn 2 m nbsp m nd2 m mm sd2 mcd2 m displaystyle m operatorname nd 2 m mm operatorname sd 2 m operatorname cd 2 m nbsp m sc2 m m nc2 dc2 m displaystyle m operatorname sc 2 m m operatorname nc 2 operatorname dc 2 m nbsp cs2 m ds2 ns2 m displaystyle operatorname cs 2 m operatorname ds 2 operatorname ns 2 m nbsp Addition theorems edit The functions satisfy the two square relations dependence on m supressed cn2 u sn2 u 1 displaystyle operatorname cn 2 u operatorname sn 2 u 1 nbsp dn2 u msn2 u 1 displaystyle operatorname dn 2 u m operatorname sn 2 u 1 nbsp From this we see that cn sn dn parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations We now may define a group law for points on this curve by the addition formulas for the Jacobi functions 3 cn x y cn x cn y sn x sn y dn x dn y 1 msn2 x sn2 y sn x y sn x cn y dn y sn y cn x dn x 1 msn2 x sn2 y dn x y, wikipedia, wiki, book, books, library,