Jacobi Elliptic Functions

The Jacobi elliptic functions are standard forms of elliptic functions. The three basic functions are denoted , , and , where is known as the elliptic modulus. They arise from the inversion of the elliptic integral of the first kind,

(1)

where , is the elliptic modulus, and is the Jacobi amplitude, giving

(2)

From this, it follows that

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

These functions are doubly periodic generalizations of the trigonometric functions satisfying

			(9)
			(10)
			(11)

In terms of Jacobi theta functions,

			(12)
			(13)
			(14)

(Whittaker and Watson 1990, p. 492), where (Whittaker and Watson 1990, p. 464) and the elliptic modulus is given by

(15)

Ratios of Jacobi elliptic functions are denoted by combining the first letter of the numerator elliptic function with the first of the denominator elliptic function. The multiplicative inverses of the elliptic functions are denoted by reversing the order of the two letters. These combinations give a total of 12 functions: cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. These functions are implemented in the Wolfram Language as JacobiSN[z, m] and so on. Similarly, the inverse Jacobi functions are implemented as InverseJacobiSN[v, m] and so on.

The Jacobi amplitude is defined in terms of by

(16)

The argument is often suppressed for brevity so, for example, can be written as .

The Jacobi elliptic functions are periodic in and as

			(17)
			(18)
			(19)

where is the complete elliptic integral of the first kind, , and (Whittaker and Watson 1990, p. 503).

The , , and functions may also be defined as solutions to the differential equations

(20)

(21)

(22)

respectively.

The standard Jacobi elliptic functions satisfy the identities

			(23)
			(24)
			(25)
			(26)

Special values include

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)

where is a complete elliptic integral of the first kind and is the complementary elliptic modulus (Whittaker and Watson 1990, pp. 498-499), and

			(33)
			(34)
			(35)

In terms of integrals,

			(36)
			(37)
			(38)
			(39)
			(40)
			(41)
			(42)
			(43)
			(44)
			(45)
			(46)
			(47)

(Whittaker and Watson 1990, p. 494).

Jacobi elliptic functions addition formulas include (where, for example, is written as for conciseness),

			(48)
			(49)
			(50)

Extended to integral periods,

			(51)
			(52)
			(53)

			(54)
			(55)
			(56)

For complex arguments,

(57)

(58)

(59)

Derivatives of the Jacobi elliptic functions include

			(60)
			(61)
			(62)

(Hille 1969, p. 66; Zwillinger 1997, p. 136).

Double-period formulas involving the Jacobi elliptic functions include

			(63)
			(64)
			(65)

Half-period formulas involving the Jacobi elliptic functions include

			(66)
			(67)
			(68)

Squared formulas include

			(69)
			(70)
			(71)

Taylor series of the Jacobi elliptic functions were considered by Hermite (1863), Schett (1977), and Dumont (1981),

			(72)
			(73)
			(74)

(Abramowitz and Stegun 1972, eqn. 16.22).