Introductions to JacobiND

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1 Introductions to JacobiND Introduction to the Jacobi elliptic functions General Historical rearks Jacobi functions are naed for the faous atheatician C. G. J. Jacobi. In 87 he introduced the elliptic aplitude a as the inverse function of the elliptic integral F by the variable and investigated the twelve functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. In the sae year, N. H. Abel independently studied properties of these functions. But earlier K. F. Gauss (799) gave soe attention to one of these function, naely sn. The odern notations for Jacobi functions were introduced later in the works of C. Guderann (838) (for functions cn, dn, and sn) and J. Glaisher (88) (for functions cd, cs, dc, ds, nc, nd, ns, sc, and sd). V. A. Puiseux (850) showed that aplitude a is a ultivalued function. Periodic functions An analytic function f is called periodic if there exists a coplex constant Ρ 0, such that f Ρ f ;. The nuber Ρ (with inial possible value of Ρ) is called the period of the function f. Exaples of well-known singly periodic functions are the exponential functions and all the trigonoetric and hyperbolic functions:, sin(), cos(), csc(), sec(), tan(), cot(), sinh(), cosh(), csch(), sech(), tanh(), and coth(), which have periods Ρ Π, Ρ Π, Ρ Π, Ρ Π, and Ρ Π. The study of such functions can be restricted to any period-strip 0 Α Ρ ; 0 Α 0, because outside this strip, the values of these functions coincide with their corresponding values inside the strip. Nonconstant analytic functions over the field of coplex nubers cannot have ore than two independent periods. So, generically, periodic functions can satisfy the following relations: f n Ρ f ; n f Ρ n Ρ f ;, n I Ρ Ρ 0,

2 where Ρ, Ρ, and Ρ are periods (basic priitive periods). The condition I Ρ Ρ 0 for doubly periodic functions iplies the existence of a period-parallelogra 0 Α Ρ Α Ρ ; 0 Α 0 Α 0, which is the analog of the periodstrip 0 Α Ρ ; 0 Α 0 for the singly periodic function with period Ρ. In the case 0 0 I Ρ Ρ 0, this parallelogra is called the basic fundaental period-parallelogra: 0,0 Α Ρ Α Ρ ; 0 Α 0 Α. The two line segents Α i Ρ i ; 0 Α i ; i, lying on the boundary of the period-parallelogra and beginning fro the origin 0 belong to 0,0. The region 0,0 includes only one corner point 0 fro four points lying at the boundary of parallelogra with corners in 0, Ρ, Ρ Ρ, Ρ. Soeties the convention 0,0 Α Ρ Α Ρ ; Α Α is used. The set of all such period-parallelogras:,n Ρ n Ρ Α Ρ Α Ρ ; 0 Α 0 Α n covers all coplex planes:,n, n n. Any doubly periodic function is called an elliptic function. The set of nubers Ρ n Ρ ;, n is called the period-lattice for the elliptic function. An elliptic function, which does not have poles in the period-parallelogra, is equal to the constant Liouville's theore. Nonconstant elliptic (doubly periodic) functions cannot be entire functions (this is not the case for singly periodic functions, for exaple, sin is an entire function. Any nonconstant elliptic function has at least two siple poles or at least one double pole in any period -parallelogra. The su of all its residues at the poles inside a period-parallelogra is ero. The nuber of eros and poles of a nonconstant elliptic function in a fundaental period-parallelogra P are finite. The nuber of eros of A, where A is any coplex nuber, in a fundaental period-parallelogra 0,0 does not depend on the value A and coincides with the nuber s of the poles b, b,, b s counted according to their ultiplicity (s is called the order of the elliptic function ). The siplest elliptic function has order two. Let a, a,, a r (and b, b,, b s ) be the eros (and poles) of a nonconstant elliptic function in a fundaental period-parallelogra 0,0, both listed one or ore ties according to their ultiplicity. This results in the following:

3 3 r s r a j b k Μ Ρ Ν Ρ ; a j 0,0 b k 0,0 a j 0 b k 0 Μ Ν. j s k So, the nuber of eros of a nonconstant elliptic function in the fundaental period-parallelogra 0,0 is equal to the nuber of poles there and counted according to their ultiplicity. The su of eros of a nonconstant elliptic function in the fundaental period-parallelogra 0,0 differs fro the su of its poles by a period Μ Ρ Ν Ρ, where Μ Ν and the values of Μ, Ν depend on the function. All elliptic functions satisfy a coon fundaental property, which generalies addition, duplication, and ultiple angle properties for the trigonoetric and hyperbolic functions such as sin, sinn ; n. It can be forulated as: n k k, which can be expressed as an algebraic function of k ; k n. In other words, there exists an irreducible polynoial Ct, t,, t n in n variables with constant coefficients, for which the following relation holds: C,,, n, k 0. n k Conversely, aong all sooth functions, only elliptic functions and their degenerations have algebraic addition theores. The siplest elliptic functions (of order two) can be divide into the following two classes: () Functions that at the period-parallelogra 0,0 have only a double pole with a residue ero (e.g., the Weierstrass elliptic functions ). () Functions that at the period-parallelogra 0,0 have only two siple poles with residues, which are equal in absolute value but opposite in sign (e.g., the Jacobian elliptic functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn ). Jacobian elliptic functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn arise as solutions to the differential equation: w Α w Β w, 3 with the following coefficients:

4 Α Β cd cn cs dc dn ds nc nd ns sc sd sn. Definitions of Jacobi functions The Jacobi elliptic aplitude a and the twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are defined by the following forulas: w a ; Fw 0 cn cosa a dn sn sina cn cd dn cn cs sn dn ds sn dc cd sc cs sd ds nc cn dn cn sn cn sn dn

5 5 nd dn ns sn. It is apparent that the aplitude function a is the inverse function to elliptic integral Fw, and the functions cn, sn, and dn are the basic Jacobi functions that are built as the cosine, sine, and derivative of the aplitude function a. The other nine Jacobi functions are the ratios of these three basic Jacobi functions or their reciprocal functions. A quick look at the Jacobi functions Here is a quick look at the graphics for the twelve Jacobi elliptic functions along the real axis for. nsx 0.5 ncx 0.5 ndx 0.5 snx 0.5 scx 0.5 sdx 0.5 cnx 0.5 csx 0.5 cdx 0.5 dnx 0.5 dsx 0.5 dcx 0.5 Connections within the group of Jacobi functions and with other elliptic functions Representations through related equivalent functions The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn can be represented through the Weierstrass siga functions: cd Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 cn Σ Σ 3 e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3

6 6 cs e e 3 Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 dc Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 dn Σ Σ 3 e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 ds e e 3 Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 nc Σ 3 Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 nd Σ 3 Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3

7 7 ns e e 3 Σ 3 Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 sc e e 3 Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 sd e e 3 Σ Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 sn e e 3 Σ 3 Σ e e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn can be represented through the Weierstrass function: cd e e 3 e e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 cn e e 3 e e e 3 e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3

8 8 cs e e 3 e e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 dc e e 3 e e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 dn e e 3 e e e 3 e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 ds e e 3 e e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 nc e e 3 e 3 e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 nd e e 3 e 3 e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3

9 9 ns e e 3 e 3 e e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 sc e e 3 e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 sd e e 3 e e 3 e ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3 snu e e 3 e e 3 e 3 ; Ω, Ω, Ω 3 Ω g, g 3, Ω g, g 3 Ω 3 g, g 3, Ω 3 g, g 3 Λ Ω 3 Ω e n Ω n n,, 3. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn can be represented through the elliptic theta functions: Π ϑ, q cd K Π ϑ 3, q K ϑ, q cn K Π Π ϑ, q K cs Π ϑ, q K Π ϑ, q K Π ϑ 3, q dc K Π ϑ, q K dn Π ϑ 3, q K Π ϑ, q K

10 0 ϑ 3, q ds K Π Π ϑ, q K nc Π ϑ, q K Π ϑ, q K nd Π ϑ, q K Π ϑ 3, q K ns Π ϑ, q K Π ϑ, q K ϑ, q sc K Π ϑ, q K Π sd ϑ Π ϑ 3, q K Π K, q sn Π ϑ, q K. Π ϑ, q K The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn can be represented through the Neville theta functions: cd ϑ c ϑ d dc ϑ d ϑ c nc ϑ n ϑ c sc ϑ s ϑ c cn ϑ c ϑ n dn ϑ d ϑ n nd ϑ n ϑ d sd ϑ s ϑ d cs ϑ c ϑ s ds ϑ d ϑ s ns ϑ n ϑ s sn ϑ s ϑ n. Relations to inverse functions The Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are connected with the corresponding inverse functions by the following forulas: af ; 3 3 Fa ;

11 cdcd cncn cscs dcdc dndn dsds ncnc ndnd nsns scsc sdsd snsn. Representations through other Jacobi functions By definition, the three basic Jacobi functions have the following representations through the aplitude function a: cn cosa dn sin a ; sn sina. The other nine Jacobi functions can be easily expressed through the three basic Jacobi functions cn, cn, and dn and, consequently, they can also be represented through the aplitude function a. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are interconnected by forulas that include rational functions, siple powers, and arithetical operations fro other Jacobi functions. These forulas can be divided into the following eleven groups: Representations of cd through other Jacobi functions are: cd cn dn cd cn nd cd cn cn cd cs ds cd cs sd cd cd dc cs cs cd cd nc dn sc ds cd dn dn cd ds cd nd nc cd nc cd nd cd nd cd ns ns

12 cd sd sc cd sc cd sd cd sn sn. Representations of cn through other Jacobi functions are: cn cd dn cn cd nd cn cd cd cn cs ns cn cs sn cn cs cs cn dn dc cn dc nd cn dc cn dn cn ds ds cn nc cn nc cn nd nd cn sc ns cn ns cn sn sc cn sc cn sd sd cn sn. Representations of cs through other Jacobi functions are: cs cd ds cs cn ns cs cd sd cs cn sn cs cs cd cd cn cn cs ds dc cs dc sd cs dc cs dn dn cs ds cs ns nc cs nc sn cs nc

13 3 cs nd nd cs ns cs ns cs sc cs sd sd cs sn cs sn sn. Representations of dc through other Jacobi functions are: dc cd dc dn cn dc ds cs dc dc nd cn sd cs dc cn cn dc cs cs dc dn nc dc dn dc dn dn dc ds sc dc ds ds dc nc nd dc nc dc nd dc nd dc ns ns dc sc sd dc sc dc sd dc sn sn. Representations of dn through other Jacobi functions are: dn cn cd dn cd nc dn cd dn dc cn dn cn dn cs cs dn dc nc dn dc dn dc dc

14 dn ds sn dn nc dn ds ns dn ds ds dn nd dn sd ns dn ns dn sc sc dn sn sd dn sd dn sn K K dn sn. Representations of ds through other Jacobi functions are: ds cs cd ds cd sc ds cd ds cn cn ds dc cs ds cs ds dc sc ds dc dc ds dn ns ds dn sn ds dn dn ds nc nc ds ns nd ds nd sn ds nd ds ns ds ds sd sc sc ds sn sn. Representations of nc through other Jacobi functions are: nc dn cd nc nd cd nc cd cd nc cn nc cn

15 5 nc ns cs nc sn cs nc cs nc dc dn nc nd dc nc dc nc nc nc dn ds ds nd nd nc ns sc nc ns ns nc sc sn nc sc nc nc sd sd sn. Representations of nd through other Jacobi functions are: nd cd cn nd cd nd nc cd nd cd nd cn dc nd cn nd cs cs nd nc dc nd dc dc nd dn nd ns ds nd sn ds nd ds ds nd nc nc nd ns sd nd ns nd sc sc ns nd sd sn nd sd nd dn nd sn nd. snk K

16 6 Representations of ns through other Jacobi functions are: ns cd cd ns cs cn ns cn sc ns cn ns nc cs ns cs ns dc dc ns ds dn ns dn sd ns dn ns nd ds ns ds ns nc sc ns nc nc ns nd sd ns nd nd ns sc ns sc sc ns sd sd ns sn. Representations of sc through other Jacobi functions are: sc ds cd sc sd cd sc cd cd sc ns cn sc sn cn sc cn cn sc cs sc dc ds sc sd dc sc dc sc sc dn dn ds sc nc ns sc sn nc sc nc sc nd nd sc ns sc ns

17 7 sc sd sd sc sn sc sn sn. Representations of sd through other Jacobi functions are: sd cd cs sd sc cd sd cd sd cn cn sd cs dc sd cs sd sc dc sd dc dc sd ns dn sd sn dn sd dn dn sd ds sd nc nc sd nd ns sd sn nd sd nd sd sd ns sc sc sd sn sn. Representations of sn through other Jacobi functions are: sn cd cd sn cn cs sn sc cn sn cn sn cs nc sn cs sn dc dc sn dn ds sn sd dn sn dn sn ds nd sn ds sn sc nc sn nc

18 8 sn sd nd sn nd nd sn ns sn sc sn sn sd sd. sc sc The best-known properties and forulas for Jacobi functions Real values for real arguents For real values of arguents and, the values of all Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are real (or infinity). Siple values at ero All thirteen Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following siple values at the origin: a0 0 0 cd0 0 cn0 0 cs0 0 dc0 0 dn0 0 ds0 0 nc0 0 nd0 0 ns0 0 sc0 0 0 sd0 0 0 sn Specific values for specialied paraeter values All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn can be represented through eleentary functions when 0 or. The twelve elliptic functions degenerate into trigonoetric and hyperbolic functions: a 0 a tan Π ak Π

19 9 cd 0 cos cd Π 0 sin cd cn 0 cos cn Π 0 sin cn sech cs 0 cot cs Π 0 tan cs csch dc 0 sec dc Π 0 csc dc dn 0 dn sech dn Π ds 0 csc ds Π nc 0 sec 0 sec ds Π csch sech nc Π 0 csc nc cosh nd 0 nd cosh nd Π sinh ns 0 csc ns Π 0 sec ns coth sc 0 tan sc Π 0 cot sc sinh sd 0 sin sd Π 0 cos sd sinh sn 0 sin sn Π 0 cos sn tanh. All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have very siple values at 0: a0 0 cd0 cn0 cs0 dc0 dn0 ds0 nc0 nd0 ns0 sc0 0 sd0 0 sn0 0. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following values at the half-quarter-period points: cd K cn K cs K dc K dn K ds K nc K nd K ns K sd K sc K sn K. The partial derivatives of all Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn at the points 0, 0, or can be represented through trigonoetric functions, for exaple:

20 0 a,0 0, a 3,0 0, a 0,, 0 sin 8 a0,, 0 6 cos 0 6 sin sin 8 a 0,, sech sinh a0,, cosh 9 sinh sech tanh 5. 3 Analyticity All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are analytical eroorphic functions of and that are defined over. Poles and essential singularities The aplitude function a does not have poles and essential singularities with respect to and. For fixed, all Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have an infinite set of singular points, including siple poles in finite points and an essential singular point. The following forulas describe the sets of the siple poles for the corresponding Jacobi functions: cd r K s K ; r, s cn r K s K ; r, s csr K s K ; r, s dc r K s K ; r, s dn r K s K ; r, s ds r K s K ; r, s ncr K s K ; r, s nd r K s K ; r, s ns r K s K ; r, s scr K s K ; r, s sd r K s K ; r, s sn r K s K ; r, s. The values of the residues of the Jacobi functions at the siple poles are given by the following forulas:

21 res cd s K r K r ; r, s res cn s K r K rs ; r, s res cs s K r K s ; r, s res dc s K r K r ; r, s res dn s K r K s ; r, s res ds s K r K rs ; r, s res nc s K r K rs ; r, s res nd s K r K s ; r, s res ns s K r K r ; r, s res sc s K r K s ; r, s res sd s K r K res sn s K r K r Branch points and branch cuts rs ; r, s. ; r, s For fixed, all Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are eroorphic functions in that have no branch points and branch cuts. For fixed, all Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn do not have branch points and branch cuts. Periodicity The Jacobi aplitude a is a pseudo-periodic function with respect to with period K and pseudoperiod K: a r K s K a r Π ; r, s 0.

22 The Jacobi functions cd, dc, ns, and sn are doubly periodic functions with respect to with periods K and K. The Jacobi functions cs, dn, nd, and sc are doubly periodic functions with respect to with periods K and K. The Jacobi functions cn, ds, nc, and sd are doubly periodic functions with respect to with periods K and K. That periodicity can be described by the following forulas: cd K cd cn K cn cs K cs dc K dc cd K cd cn K cn cs K cs dc K dc dn K dn dn K dn ds K ds ds K ds nc K nc nc K nc nd K nd nd K nd ns K ns ns K ns sc K sc sc K sc sd K sd sd K sd sn K sn sn K sn. The periodicity of Jacobi functions follow fro ore general forulas that also describe quasi-periodicity situations such as sn K sn : cd r K s K r cd ; r, s cn s K r K rs cn ; r, s cs s K r K s cs ; r, s dc s K r K r dc ; r, s dn s K r K s dn ; r, s ds s K r K rs ds ; r, s nc s K r K rs nc ; r, s nd s K r K s nd ; r, s ns s K r K r ns ; r, s sc s K r K s sc ; r, s sd s K r K rs sd ; r, s

23 3 sn s K r K r sn ; r, s. Parity and syetry All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have irror syetry: a a cd cd cn cn cs cs dc dc dn dn ds ds nc nc nd nd ns ns sc sc sd sd sn sn. The Jacobi functions cd, cn, dc, dn, nc, and nd are even functions with respect to : cd cd dc dc nc nc cn cn dn dn nd nd. The Jacobi functions a, cs, ds, ns, sc, sd, and sn are odd functions with respect to : a a cs cs ds ds ns ns sd sd sc sc sn sn. Series representations The Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following series expansions at the point 0: a ; 0 cd 6 5 ; 0 cn ; 0 cs ; 0 dc 5 6 ; 0

24 dn ; 0 ds ; 0 nc 5 ; 0 nd 5 ; 0 ns ; 0 sc ; 0 sd ; 0 sn 5 3 ; The Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following series expansions at the point 0: a sin cos 0 6 sin sin ; 0 cd cos sin cos sin cos 8 cos3 cos5 sin sin3 ; 0 cn cos sin sin cos 8 cos3 cos5 6 sin sin3 ; 0 cs cot csc cot cos 3 cos3 cos5 8 sin csc 3 ; 0 dc sec cos sin tan sec 5 8 cos sin sin 5 cos 5 sec 3 ; 0 dn sin 3 sin 8 cos 5 sin sin3 ; 0

25 5 ds csc 6 cos 7 sin sin3 csc 5 sin sin 8 cos 3 cos 9 csc 3 ; 0 nc sec sin sec tan 5 8 cos sin sin cos sec 3 ; nd sin 3 sin 8 cos sin sin3 ; 0 ns csc cot cos sin csc 5 8 cos sin sin 5 cos 5 csc 3 ; 0 sc tan tan sec cos 6 8 cos cos 9 sin sec 3 ; sd sin cos 7 sin sin cos cos sin 6 sin3 sin5 ; 0 sn sin cos sin 8 56 cos cos3 9 cos cos 8 sin ; 0. The Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following series expansions at the point : a tan Π sech sinh 6 cosh 9 sinh sech tanh 5 ; cd sinh 3 sinh 8 cosh sinh sinh3 ; cn sech sinh sech tanh 5 8 cosh sinh sinh cosh sech 3 ;

26 6 cs csch coth cosh csch 5 8 cosh sinh sinh 5 cosh 5 csch 3 ; dc sinh 3 sinh 8 cosh 5 sinh sinh3 ; dn sech tanh cosh sinh sech 5 8 cosh sinh sinh 5 cosh 5 sech 3 ; ds csch 6 cosh 7 sinh sinh3 csch 5 sinh sinh 8 cosh 3 cosh 9 csch 3 ; nc cosh sinh sinh cosh 8 cosh3 cosh5 6 sinh sinh3 ; nd cosh sinh cosh sinh cosh 8 cosh3 cosh5 sinh sinh3 ; ns coth coth csch cosh 3 cosh3 cosh5 8 sinh csch 3 ; sc sinh cosh sinh 8 56 cosh cosh3 8 7 sinh 8 sinh3 sinh5 ; sd sinh cosh 7 sinh sinh cosh cosh3 5 cosh cosh 3 sinh ; sn tanh sech tanh 5 7 cosh 8 5 sinh 9 sinh3 sinh5 sech 3 ;. q-series representations

27 7 The Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following so-called q-series representations: a Π K q k sin k Π k k q k K cd Π k q k cos K k 0 q k k Π K cn Π q n cos n Π K n 0 q n K cs dc Π K cot Π K Π K sec Π K Π K k q k q k sin k Π K Π K k q k cos k 0 q k k Π K dn Π K Π K n q n q n cos n Π K ds Π K csc Π K Π K sin k 0 q k q k k Π K Π nc sec K Π K Π k q k cos K k 0 q k k Π K Π nd K ns Π K csc Π K Π k q k K k q k Π K k 0 q k cos k Π K k Π sin k q K Π sc tan K Π K Π k q k K k q k sin k Π K sd Π k q k sin K k 0 q k k Π K sn Π q n sin n Π K n 0 q n K, where q is the elliptic noe and K is the coplete elliptic integral. Product representations

28 8 The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn have the following product representations: cd q cos Π K k q k cos Π k q K q k cos Π k q K cn q cos Π K n q n cos Π n q K q n cos Π n q K cs cot Π K k q k cos Π k q K q k cos Π k q K dc q sec Π K k q k cos Π k q K q k cos Π k q K dn n q n cos Π n q K q n cos Π n q K ds q csc Π K k q k cos Π k q K q k cos Π k q K nc q sec Π K n q n cos Π n q K q n cos Π n q K nd n q n cos Π n q K q n cos Π n q K ns q csc Π K k q k cos Π k q K q k cos Π k q K sc tan Π K k q k cos Π k q K q k cos Π k q K sd q sin Π K k q k cos Π k q K q k cos Π k q K sn q sin Π K k q k cos Π k q K, q k cos Π K q k

29 9 where q is the elliptic noe and K is the coplete elliptic integral. Transforations The aplitude function a satisfies nuerous relations that allow for transforations of its arguents, for exaple: a Π ak ;, a Π ak ;. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn with specific arguents can soeties be represented through elliptic functions with other ostly sipler arguents, for exaple: cd nd dc dn nc cn cn cn dn dn cn nd nd nc cs, ns, ds ds ns cs sc, sn, sd sd sn sn cn cd nd cn cn dc dn nc cn dn nd dn cn nd nc cs ns ds ds ns cs sc sn sd sd sn sn cn cd dc nc sc cd dc nc dc cn dn cs cd dn cn ds cs nd nd nc ns sd sd sc sn cn cn cn dn nc dn dn nc nd nd cs ds ds ns sn ns dn ns cs ds sc sc sd sn

30 30 sn sn dn. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn with the arguent coplex u v can be represented through elliptic functions with arguents u and v, for exaple: cnu cnv snu dnu snv dnv cdu v dnu dnv snu cnu snv cnv cnu cnv snu dnu snv dnv cnu v snu snv cnu cnv snu dnu snv dnv csu v cnv dnv snu cnu dnu snv dnu dnv snu cnu snv cnv dcu v cnu cnv snu dnu snv dnv dnu dnv snu cnu snv cnv dna v snu snv dnu dnv snu cnu snv cnv dsu v cnv dnv snu cnu dnu snv snu snv ncu v cnu cnv snu dnu snv dnv snu snv ndu v dnu dnv snu cnu snv cnv snu snv nsu v cnv dnv snu cnu dnu snv cnv dnv snu cnu dnu snv scu v cnu cnv snu dnu snv dnv cnv dnv snu cnu dnu snv sdu v dnu dnv snu cnu snv cnv snu v cnu dnu snv cnv dnv snu snu snv. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn satisfy the following half-angle forulas:

31 3 cd dc nc sc cndn dn cn dn cn cndn dn cndn cn cndn cn cndn dn dn nd sd dn cn dn dn dn cn cn dn cn cs cndn cn ds ns dn cn dn cn cn sn cn dn. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn satisfy the following double-angle (or ultiplication) forulas: cd cn sn dn dn sn cn cn cn sn dn sn cnn n n K Μ Ν Τ cn n Μ,Ν 0 ; n cs cn sn dn sn cn dn dc dn sn cn cn sn dn dn dn sn cn sn dnn n n K Μ Ν Τ dn n Μ,Ν 0 ; n ds dn sn cn sn cn dn sn nc cn sn dn ncn n n K Μ Ν Τ nc n Μ,Ν 0 ; n sn nd dn sn cn ndn n n Μ,Ν 0 K Μ Ν Τ nd n ; n

32 3 sn ns sn cn dn nsn, n n n Μ n n Ν n Μ K Ν K ns n ; n sn cn dn sc cn sn dn sn cn dn sd dn sn cn sn cn dn sn sn snn, n n n Μ n n Ν n Μ K Ν K sn n ; n. Identities The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn satisfy the following nonlinear functional equations: w w w w w 0 ; w cd w w w w w 0 ; w cn w w w w w 0 ; w cs w w w w w 0 ; w dc w w w w w 0 ; w dn w w w w w 0 ; w ds w w w w w 0 ; w nc w w w w w 0 ; w nd w w w w w 0 ; w ns w w w w w 0 ; w sc w w w w w 0 ; w sd w w w w w 0 ; w sn.

33 33 Siple representations of derivatives The derivatives of all Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn with respect to variable have rather siple and syetrical representations that can be expressed through other Jacobi functions: a dn cd nd sd cn sn dn cs ds ns dc nc sc dn sn cn ds cs ns nc dc sc nd cd sd ns, cs ds sc dc nc sd cd nd sn cn dn. The derivatives of all Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn with respect to variable have ore coplicated representations that include other Jacobi functions and the elliptic integral Ea :

34 3 a Ea dn cn sn ; 3 3 cd Ea nd sd cn sn dn Ea sn cd cs ns ds Ea sn cd dc sc nc Ea dn sn cn Ea dn sc ds cs ns Ea dn sc nc sc dc Ea cd sn nd sd cd Ea dn sc ns ds cs Ea sn cd sc nc dc Ea cd sn sd cd nd Ea dn sc sn dn cn Ea cd sn. Integration The indefinite integrals of the twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn with respect to variable can be expressed through Jacobi and eleentary functions by the following forulas: lognd sd cd

35 35 cn cos dn sn dn cs logns ds dc lognc sc dn a ds log cn sn logdc sc nc nd cd cos cd sd ns logds cs logdc nc sc sin cd cd dn sd sn logdn cn. Differential equations The Jacobi aplitude a satisfies the following differential equations: w dn 0 ; w a w 3 w w 5 w 3 w w w 0 ; w a. All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are special solutions of ordinary second-order nonlinear differential equations: w w w 0 ; w a w w w 0 ; w cd

36 36 w w w 0 ; w cn w w w 0 ; w cs w w w 0 ; w dc w w w 0 ; w dn w w w 0 ; w ds w w w 0 ; w nc w w w 0 ; w nd w w w 0 ; w ns w w w 0 ; w sc w w w 0 ; w sd w w w 0 ; w sn. The twelve Jacobi functions cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn satisfy very coplicated ordinary differential equations with respect to variable, for exaple: 3 w w w w w 3 3 w 6 w w 6 w w 6 w w w 3 3 w w 3 6 w 8 8 w w w 6 w 6 w w 6 6 w 3 w 6 w 8 w w 7 w w 3 8 w w 8 w w 7 w w 3 8 w 6 w w 5 3 w 3 3 w w 3 w w w w 3 w w ; w sn. Zeros All Jacobi functions a, cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn are equal to ero in the points k K n K, where K is the coplete elliptic integral of the first kind and k, n are even or odd integers: a s K 0 ; s

37 37 cd r K s K 0 ; r, s cnr s K s K 0 ; r, s cs r K s K 0 ; r, s dc r K s K 0 ; r, s dn r K s K 0 ; r, s ds r K s K 0 ; r, s nc r K s K 0 ; r, s nd r K s K 0 ; r, s ns r K s K 0 ; r, s sc r K s K 0 ; r, s sd r K s K 0 ; r, s sn r K s K 0 ; r, s. Applications of Jacobi elliptic functions Applications of Jacobi elliptic functions include conforal appings, electrostatics and agnetostatics, fluid dynaics, echanics of tops, nonlinear integrable equations, the Ising odel of statistical echanics, celestial echanics, closed-for solutions for nonlinear Schr ödinger equations, analysis of exactly solvable chaos-exhibiting sequences, solution of equations of otion for quartic potentials, and elliptic function theory.

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Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values InverseJacobiCS Notations Traditional nae Inverse of the Jacobi elliptic function cs Traditional notation cs Matheatica StandardFor notation InverseJacobiCS, Priary definition 09.39.0.000.0 csw ; w cs