Jacobi Theta Functions

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted  in modern texts, although the notations  and  (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.

The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].

The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).

The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where  and  and  are related by

(1)

Let the multivalued function  be interpreted to stand for . Then for a complex number , the Jacobi theta functions are defined as

			(2)
			(3)
			(4)
			(5)

Writing the doubly infinite sums as singly infinite sums gives the slightly less symmetrical forms

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

(Whittaker and Watson 1990, pp. 463-464). Explicitly writing out the series gives

			(12)
			(13)
			(14)
			(15)

(Borwein and Borwein 1987, p. 52; Whittaker and Watson 1990, p. 464).  is an odd function of , while the other three are even functions of .

The following table illustrates the quasi-double periodicity of the Jacobi theta functions.

	1
	1

Here,

(16)

The quasi-periodicity can be established as follows for the specific case of ,

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)

The Jacobi theta functions can be written in terms of each other:

			(25)
			(26)
			(27)

(Whittaker and Watson 1990, p. 464). Any Jacobi theta function of given arguments can be expressed in terms of any other two Jacobi theta functions with the same arguments.

The functions  and  satisfy the identity

(28)

Define

(29)

to be the Jacobi theta functions with argument , plotted above. Then the doubly infinite sums (◇) to (◇) take on the particularly simple forms

			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)

(OEIS A089800, A000122, and A002448; Borwein and Borwein 1987, p. 33).

The function  is also given by

(37)

where  is a q-Pochhammer symbol.

The function

			(38)
			(39)
			(40)

is sometimes defined in number theoretic contexts (Davenport 1980, p. 62). Similarly, the function

			(41)
			(42)

is sometimes also defined (Edwards 2001, p. 15). This function satisfies

(43)

(Jacobi 1828; Riemann 1859; Edwards 2001, p. 15), which Jacobi attributes to Poisson and follows from the Poisson sum formula. Is also satisfies the identity

(44)

(Edwards 2001, p. 17).

Special values include

(45)

and

(46)

where  is the gamma function, most which are all special cases of the Ramanujan theta functions.

A special derivative value due to O. Marichev (pers. comm., Jul. 2008) is given by

(47)

The plots above show the Jacobi theta functions plotted as a function of argument  and nome  restricted to real values.

Particularly beautiful plots are obtained by examining the real and imaginary parts of  for fixed  in the complex plane for , illustrated above.

The Jacobi theta functions satisfy an almost bewilderingly large number of identities involving the four functions, their derivatives, multiples of their arguments, and sums of their arguments. Among the unusual identities given by Whittaker and Watson (1990) are

			(48)
			(49)

(Whittaker and Watson 1990, p. 464) and

			(50)
			(51)

(Whittaker and Watson 1990, p. 465), for , ..., 4, where  and . A class of identities involving the squares of Jacobi theta functions are

			(52)
			(53)
			(54)
			(55)

(Whittaker and Watson 1990, p. 466). Taking  in (55) gives the special case

(56)

which is the only identity of this type.

In addition,

			(57)
			(58)

The Jacobi theta functions obey addition rules such as

			(59)
			(60)
			(61)
			(62)
			(63)
			(64)
			(65)
			(66)

(Whittaker and Watson 1990, p. 487),

			(67)
			(68)
			(69)
			(70)
			(71)
			(72)
			(73)
			(74)

(Whittaker and Watson 1990, p. 488), and

(75)

(Whittaker and Watson 1990, p. 488).

There are also a series of duplication formulas:

			(76)
			(77)
			(78)
			(79)
			(80)
			(81)

(Whittaker and Watson 1990, p. 488).

Ratios of Jacobi theta function derivatives to the functions themselves have the simple forms

			(82)
			(83)
			(84)
			(85)
			(86)
			(87)
			(88)
			(89)

(Whittaker and Watson 1990, p. 489).

The Jacobi theta functions can be expressed as products instead of sums by

			(90)
			(91)
			(92)
			(93)

where

(94)

(Whittaker and Watson 1990, pp. 469-470).

Additional beautiful product ("Eulerian") forms are given by Zucker (1990), partially summarized in the following table, where

(95)

and the q-products are written , , , and .

theta function	Sloane	Eulerian	Jacobian
	A000122
	A002448
	A089798
	A089799
	A089800
	A089801
	A089802
	A089805
	A080995
	A089806
	A089807
	A089810
	A089811
	A089812
	A089813

Additional identities include

			(96)
			(97)

Here,

(98)

(OEIS A022597).

The Jacobi theta functions satisfy the partial differential equation

(99)

where . Ratios of the Jacobi theta functions with  in the denominator also satisfy differential equations

(100)

(101)

(102)

Jacobi's imaginary transformation expresses  in terms of . There are a large number of beautiful identities involving Jacobi theta functions of arguments , , , and  and , , , and , related by

			(103)
			(104)
			(105)
			(106)

(Whittaker and Watson 1990, pp. 467-469, 488, and 490). Using the notation

(107)

(108)

gives a whopping 288 identities of the form

(109)

The complete elliptic integrals of the first and second kinds can be expressed using Jacobi theta functions. Let

(110)

and plug into (◇)

(111)

Now write

(112)

and

(113)

Then

(114)

where the elliptic modulus is defined by

(115)

Define also the complementary elliptic modulus

(116)

Now, since

(117)

we have shown

(118)

The solution to the equation is

(119)

which is a Jacobi elliptic function with periods

(120)

and

(121)

Letting  be the complete elliptic integral of the first kind with modulus , then

			(122)
			(123)
			(124)

where  is the complementary modulus.

The Jacobi theta functions provide analytic solutions to many tricky problems in mathematics and mathematical physics. For example, the Jacobi theta functions are related to the sum of squares function  giving the number of representations of  by two squares via

			(125)
			(126)

(Borwein and Borwein 1987, p. 34). The general quintic equation is solvable in terms of Jacobi theta functions, and these functions also provide a uniformly convergent form of the Green's function for a rectangular region (Oberhettinger and Magnus 1949).

Finally, Jacobi theta functions can be used to uniformize all elliptic curves. Jacobi elliptic functions may also be used to uniformize some hyperelliptic curves, although only two such examples are known. The classical example is the Burnside curve, and the second was found by Farkas and Kra around 1995.

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