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Jacobi Theta Functions

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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 theta_n(z,q) in modern texts, although the notations Theta_n(z,q) and theta_n(z,q) (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 q, denoted theta_n(z,q), or the half-period ratio tau, denoted theta_n(z|tau), where |q|<1 and q and tau are related by

q=e^(ipitau).
(1)

Let the multivalued function q^lambda be interpreted to stand for e^(lambdapiitau). Then for a complex number z, the Jacobi theta functions are defined as

theta_1(z,q)=sum_(n=-infty)^(infty)(-1)^(n-1/2)q^((n+1/2)^2)e^((2n+1)iz)
(2)
theta_2(z,q)=sum_(n=-infty)^(infty)q^((n+1/2)^2)e^((2n+1)iz)
(3)
theta_3(z,q)=sum_(n=-infty)^(infty)q^(n^2)e^(2niz)
(4)
theta_4(z,q)=sum_(n=-infty)^(infty)(-1)^nq^(n^2)e^(2niz).
(5)

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

theta_1(z,q)=2sum_(n=0)^(infty)(-1)^nq^((n+1/2)^2)sin[(2n+1)z]
(6)
=2q^(1/4)sum_(n=0)^(infty)(-1)^nq^(n(n+1))sin[(2n+1)z]
(7)
theta_2(z,q)=2sum_(n=0)^(infty)q^((n+1/2)^2)cos[(2n+1)z]
(8)
=2q^(1/4)sum_(n=0)^(infty)q^(n(n+1))cos[(2n+1)z]
(9)
theta_3(z,q)=1+2sum_(n=1)^(infty)q^(n^2)cos(2nz)
(10)
theta_4(z,q)=1+2sum_(n=1)^(infty)(-1)^nq^(n^2)cos(2nz)
(11)

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

theta_1(z,q)=2q^(1/4)sinz-2q^(9/4)sin(3z)+2q^(25/4)sin(5z)+...
(12)
theta_2(z,q)=2q^(1/4)cosz+2q^(9/4)cos(3z)+2q^(25/4)cos(5z)+...
(13)
theta_3(z,q)=1+2qcos(2z)+2q^4cos(4z)+2q^9cos(6z)+...
(14)
theta_4(z,q)=1-2qcos(2z)+2q^4cos(4z)-2q^9cos(6z)+...
(15)

(Borwein and Borwein 1987, p. 52; Whittaker and Watson 1990, p. 464). theta_1(z,q) is an odd function of z, while the other three are even functions of z.

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

theta_itheta_i(z+pi)/theta_i(z)theta_i(z+taupi)/theta_i(z)
theta_1-1-N
theta_2-1N
theta_31N
theta_41-N

Here,

N=q^(-1)e^(-2iz).
(16)

The quasi-periodicity can be established as follows for the specific case of theta_4,

theta_4(z+pi,q)=sum_(n=-infty)^(infty)(-1)^nq^(n^2)e^(2niz)e^(2nipi)
(17)
=sum_(n=-infty)^(infty)(-1)^nq^(n^2)e^(2niz)
(18)
=theta_4(z,q)
(19)
theta_4(z+pitau,q)=sum_(n=-infty)^(infty)(-1)^nq^(n^2)e^(2nipit)e^(2niz)
(20)
=sum_(n=-infty)^(infty)(-1)^nq^(n^2)q^(2n)e^(2niz)
(21)
=-q^(-1)e^(-2iz)sum_(n=-infty)^(infty)(-1)^(n+1)q^((n+1)^2)q^(2(n+1)iz)
(22)
=-q^(-1)e^(-2iz)sum_(n=-infty)^(infty)(-1)^nq^(n^2)q^(2niz)
(23)
=-q^(-1)e^(-2iz)theta_4(z,q).
(24)

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

theta_1(z,q)=-ie^(iz+piitau/4)theta_4(z+1/2pitau,q)
(25)
theta_2(z,q)=theta_1(z+1/2pi,q)
(26)
theta_3(z,q)=theta_4(z+1/2pi,q)
(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 theta_3(z,q) and theta_4(z,q) satisfy the identity

theta_4(z,q)=theta_3(z,-q).
(28)
JacobiThetaFunctions

Define

theta_i(q)=theta_i(z=0,q)
(29)

to be the Jacobi theta functions with argument z=0, plotted above. Then the doubly infinite sums (◇) to (◇) take on the particularly simple forms

theta_1(q)=0
(30)
theta_2(q)=sum_(n=-infty)^(infty)q^((n+1/2)^2)
(31)
=q^(1/4)(2+2q^2+2q^6+2q^(12)+2q^(20)+2q^(30)+...)
(32)
theta_3(q)=sum_(n=-infty)^(infty)q^(n^2)
(33)
=1+2q+2q^4+2q^9+2q^(16)+2q^(25)+...
(34)
theta_4(q)=sum_(n=-infty)^(infty)(-1)^nq^(n^2)
(35)
=1-2q+2q^4-2q^9+2q^(16)-2q^(25)+...
(36)

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

The function theta_3(q) is also given by

theta_3(q)=((-q,-q)_infty)/((q,-q)_infty),
(37)

where (a;q)_infty is a q-Pochhammer symbol.

The function

theta(x)=sum_(n=-infty)^(infty)e^(-n^2pix)
(38)
=theta_3(0,e^(-pix))
(39)
=theta_3(0|ix)
(40)

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

psi(x)=sum_(n=1)^(infty)e^(-n^2pix)
(41)
=1/2[theta_3(0,e^(-pix))-1]
(42)

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

(1+2psi(x))/(1+2psi(x^(-1)))=1/(sqrt(x))
(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

1/2+psi(1)+4psi^'(1)=0
(44)

(Edwards 2001, p. 17).

Special values include

theta_3(e^(-pi))=(pi^(1/4))/(Gamma(3/4)) theta_3(e^(-pisqrt(2)))=(Gamma(9/8))/(Gamma(5/4))sqrt((Gamma(1/4))/(2^(1/4)pi)) theta_3(e^(-pisqrt(6))) =[-(Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24)))/(16sqrt(6)(-18-12sqrt(2)+10sqrt(3)+7sqrt(6))pi^3)]^(1/4) theta_4(-e^(-pi))=(pi^(1/4))/(Gamma(3/4)) theta_4(e^(-pi))=(pi^(1/4))/(2^(1/4)Gamma(3/4))
(45)

and

(theta_2(-e^(-pisqrt(3))))/(theta_3(-e^(-pisqrt(3))))=(4sqrt(3)-7)^(1/4),
(46)

where Gamma(z) 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

theta_4^'(e^(-pi))=-(e^pi[pi^2-2Gamma^4(3/4)])/(8·2^(1/4)pi^(3/4)Gamma^5(3/4)).
(47)
JacobiThetaZQ

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

JacobiTheta1
JacobiTheta2
JacobiTheta3
JacobiTheta4

Particularly beautiful plots are obtained by examining the real and imaginary parts of theta_i(z,q) for fixed z in the complex plane for |q|<1, 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

theta_3(z,q)=theta_3(2z,q^4)+theta_2(2z,q^4)
(48)
theta_4(z,q)=theta_3(2z,q^4)-theta_2(2z,q^4)
(49)

(Whittaker and Watson 1990, p. 464) and

(theta_k^'(z+pi))/(theta_k(z+pi))=(theta_k^'(z))/(theta_k(z))
(50)
(theta_k^'(z+pitau))/(theta_k(z+pitau))=-2i+(theta_k^'(z))/(theta_k(z))
(51)

(Whittaker and Watson 1990, p. 465), for k=1, ..., 4, where theta_k(z)=theta_k(z,q) and theta_i=theta_i(0,q). A class of identities involving the squares of Jacobi theta functions are

theta_1^2(z)theta_4^2=theta_3^2(z)theta_2^2-theta_2^2(z)theta_3^2
(52)
theta_2^2(z)theta_4^2=theta_4^2(z)theta_2^2-theta_1^2(z)theta_3^2
(53)
theta_3^2(z)theta_4^2=theta_4^2(z)theta_3^2-theta_1^2(z)theta_2^2
(54)
theta_4^2(z)theta_4^2=theta_3^2(z)theta_3^2-theta_2^2(z)theta_2^2
(55)

(Whittaker and Watson 1990, p. 466). Taking z=0 in (55) gives the special case

theta_4^4=theta_3^4-theta_2^4,
(56)

which is the only identity of this type.

In addition,

theta_3^2(x)=1+4(x/(1-x)-(x^3)/(1-x^3)+(x^5)/(1-x^5)-(x^7)/(1-x^7)+...)
(57)
theta_3^4(x)=1+8(x/(1-x)+(2x^2)/(1+x^2)+(3x^3)/(1-x^3)+(4x^4)/(1+x^4)+...).
(58)

The Jacobi theta functions obey addition rules such as

theta_1(y+z)theta_1(y-z)theta_4^2=theta_3^2(y)theta_2^2(z)-theta_2^2(y)theta_3^2(z)
(59)
=theta_1^2(y)theta_4^2(z)-theta_4^2(y)theta_1^2(z)
(60)
theta_2(y+z)theta_2(y-z)theta_4^2=theta_4^2(y)theta_2^2(z)-theta_1^2(y)theta_3^2(z)
(61)
=theta_2^2(y)theta_4^2(y)-theta_3^2(y)theta_1^2(z)
(62)
theta_3(y+z)theta_3(y-z)theta_4^2=theta_4^2(y)theta_3^2(z)-theta_1^2(y)theta_2^2(z)
(63)
=theta_3^2(y)theta_4^2(z)-theta_2^2(y)theta_1^2(z)
(64)
theta_4(y+z)theta_4(y-z)theta_4^2=theta_3^2(y)theta_3^2(z)-theta_2^2(y)theta_2^2(z)
(65)
=theta_4^2(y)theta_4^2(z)-theta_1^2(y)theta_1^2(z)
(66)

(Whittaker and Watson 1990, p. 487),

theta_3(y+z)theta_3(y-z)theta_2^2=theta_3^2(y)theta_2^2(z)+theta_4^2(y)theta_1^2(z)
(67)
=theta_2^2(y)theta_3^2(z)+theta_1^2(y)theta_4^2(z)
(68)
theta_3(y+z)theta_3(y-z)theta_3^2=theta_1^2(y)theta_1^2(z)+theta_3^2(y)theta_3^2(z)
(69)
=theta_2^2(y)theta_2^2(z)+theta_4^2(y)theta_4^2(z)
(70)
theta_4(y+z)theta_4(y-z)theta_2^2=theta_4^2(y)theta_2^2(z)+theta_3^2(y)theta_1^2(z)
(71)
=theta_2^2(y)theta_4^2(z)+theta_1^2(y)theta_3^2(z)
(72)
theta_4(y+z)theta_4(y-z)theta_3^2=theta_4^2(y)theta_3^2(z)+theta_2^2(y)theta_1^2(z)
(73)
=theta_3^2(y)theta_4^2(z)+theta_1^2(y)theta_2^2(z)
(74)

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

theta_1(y+/-z)theta_2(y∓z)theta_3theta_4 =theta_1(y)theta_2(y)theta_3(z)theta_4(z)+/-theta_3(y)theta_4(y)theta_1(z)theta_2(z) theta_1(y+/-z)theta_3(y∓z)theta_2theta_4 =theta_1(y)theta_3(y)theta_2(z)theta_4(z)+/-theta_2(y)theta_4(y)theta_1(z)theta_3(z) theta_1(y+/-z)theta_4(y∓z)theta_2theta_3 =theta_1(y)theta_4(y)theta_2(z)theta_3(z)+/-theta_2(y)theta_3(y)theta_1(z)theta_4(z) theta_2(y+/-z)theta_3(y∓z)theta_2theta_3 =theta_2(y)theta_3(y)theta_2(z)theta_3(z)∓theta_1(y)theta_4(y)theta_1(z)theta_4(z) theta_2(y+/-z)theta_4(y∓z)theta_2theta_4 =theta_2(y)theta_4(y)theta_2(z)theta_4(z)∓theta_1(y)theta_3(y)theta_1(z)theta_3(z) theta_3(y+/-z)theta_4(y∓z)theta_3theta_4 =theta_3(y)theta_4(y)theta_3(z)theta_4(z)∓theta_1(y)theta_2(y)theta_1(z)theta_2(z)
(75)

(Whittaker and Watson 1990, p. 488).

There are also a series of duplication formulas:

theta_3(2z)theta_3^3=theta_3^4(z)+theta_1^4(z)
(76)
theta_2(2z)theta_2theta_4^2=theta_2^2(z)theta_4^2(z)-theta_1^2(z)theta_3^2(z)
(77)
theta_3(2z)theta_3theta_4^2=theta_3^2(z)theta_4^2(z)-theta_1^2(z)theta_2^2(z)
(78)
theta_4(2z)theta_4^3=theta_3^4(z)-theta_2^4(z)
(79)
=theta_4^4(z)-theta_1^4(z)
(80)
theta_1(2z)theta_2theta_3theta_4=2theta_1(z)theta_2(z)theta_3(z)theta_4(z)
(81)

(Whittaker and Watson 1990, p. 488).

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

(theta_1^'(z))/(theta_1(z))=cotz+4sum_(n=1)^(infty)(q^(2n))/(1-q^(2n))sin(2nz)
(82)
=cotz+4sum_(n=1)^(infty)(q^(2n)sin(2z))/(q^(4n)-2q^(2n)cos(2z)+1)
(83)
(theta_2^'(z))/(theta_2(z))=-tanz+4sum_(n=1)^(infty)(-1)^n(q^(2n))/(1-q^(2n))sin(2nz)
(84)
=-tanz-4sum_(n=1)^(infty)(q^(2n)sin(2z))/(q^(4n)+2q^(2n)cos(2z)+1)
(85)
(theta_3^'(z))/(theta_3(z))=4sum_(n=1)^(infty)(-1)^n(q^n)/(1-q^(2n))sin(2nz)
(86)
=-4sum_(n=1)^(infty)(q^(2n-1)sin(2z))/(2q^(2n-1)cos(2z)+q^(4n-2)+1)
(87)
(theta_4^'(z))/(theta_4(z))=4sum_(n=1)^(infty)(q^(2n-1)sin(2z))/(1-2q^(2n-1)cos(2z)+q^(4n-2))
(88)
=4sum_(n=1)^(infty)(q^nsin(2nz))/(1-q^(2n))
(89)

(Whittaker and Watson 1990, p. 489).

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

theta_1(z)=2Gq^(1/4)sinzproduct_(n=1)^(infty)[1-2q^(2n)cos(2z)+q^(4n)]
(90)
theta_2(z)=2Gq^(1/4)coszproduct_(n=1)^(infty)[1+2q^(2n)cos(2z)+q^(4n)]
(91)
theta_3(z)=Gproduct_(n=1)^(infty)[1+2q^(2n-1)cos(2z)+q^(4n-2)]
(92)
theta_4(z)=Gproduct_(n=1)^(infty)[1-2q^(2n-1)cos(2z)+q^(4n-2)],
(93)

where

G=product_(n=1)^infty(1-q^(2n))
(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

(n)=product_(k=1)^infty(1-q^(kn))
(95)

and the q-products are written w=Q_0, x=Q_1, y=Q_2, and z=Q_3.

theta functionOEISEulerianJacobian
theta_3(q)A000122((2)^5)/((1)^2(4)^2)wz^2
theta_4(q)A002448((1)^2)/((2))wy^2
theta_4(q^2)A089798((2)^2)/((4))wyz
theta_2(q^(1/2))A0897992q^(1/8)((2)^2)/((1))2q^(1/8)wxz
theta_2(q)A0898002q^(1/4)((4)^2)/((2))2q^(1/4)wx^2
1/2[theta_3(q^(1/3))-theta_3(q^3)]A089801q^(1/3)((2)^2(3)(12))/((1)(4)(6))q^(1/3)z|wxy|q^3
1/2[theta_4(q^3)-theta_4(q^(1/3))]A089802q^(1/3)((1)(6)^2)/((2)(3))q^(1/3)y|wxz|q^3
1/2[theta_4(q^6)-theta_4(q^(2/3))]A089805q^(2/3)((2)(12)^2)/((4)(6))q^(2/3)yz|wx^2|q^3
1/2[theta_2(q^(1/6))-theta_2(q^(3/2))]A080995q^(1/24)((2)(3)^2)/((1)(6))q^(1/24)xz|wy^2|q^3
1/2[theta_2(q^(1/3))-theta_2(q^3)]A089806q^(1/12)((4)(6)^2)/((2)(12))q^(1/12)x|wyz|q^3
1/2[3theta_3(q^9)-theta_3(q)]A089807((1)(4)(6)^2)/((2)(3)(12))wxy|z|q^3
1/2[3theta_4(q^9)-theta_4(q)]A089810((2)^2(3))/((1)(6))wxz|y|q^3
1/2[3theta_4(q^(18))-theta_4(q^2)]A089811((4)^2(6))/((2)(12))wx^2|yz|q^3
1/2[theta_2(q^(1/2))-3theta_2(q^(9/2))]A089812q^(1/8)((1)^2(6))/((2)(3))q^(1/8)wy^2|xz|q^3
1/2[theta_2(q)-3theta_2(q^9)]A089813q^(1/4)((2)^2(12))/((4)(6))q^(1/4)wyz|x|q^3

Additional identities include

theta_4(q)=(2)product_(k=1)^(infty)(1-q^(2k-1))^2
(96)
theta_4^3(q)=((1)^4)/((2))product_(k=1)^(infty)(1-q^(2k-1))^2.
(97)

Here,

product_(k=1)^infty(1-q^(2k-1))^2=1-2q+q^2-2q^3+4q^4+...
(98)

(OEIS A022597).

The Jacobi theta functions satisfy the partial differential equation

1/4pii(partial^2y)/(partialz^2)+(partialy)/(partialtau)=0,
(99)

where y=theta_j(z|tau). Ratios of the Jacobi theta functions with theta_4 in the denominator also satisfy differential equations

d/(dz)[(theta_1(z))/(theta_4(z))]=theta_4^2(theta_2(z)theta_3(z))/(theta_4^2(z))
(100)
d/(dz)[(theta_2(z))/(theta_4(z))]=-theta_3^2(theta_1(z)theta_3(z))/(theta_4^2(z))
(101)
d/(dz)[(theta_3(z))/(theta_4(z))]=-theta_2^2(theta_1(z)theta_2(z))/(theta_4^2(z)).
(102)

Jacobi's imaginary transformation expresses theta_i(z/tau|-1/tau) in terms of theta_i(z|tau). There are a large number of beautiful identities involving Jacobi theta functions of arguments w, x, y, and z and w^', x^', y^', and z^', related by

2w^'=-w+x+y+z
(103)
2x^'=w-x-y+z
(104)
2y^'=w+x-y+z
(105)
2z^'=w+x+y-z
(106)

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

theta_i(w+pi/2,q)theta_j(x+pi/2,q)theta_k(y,q)theta_l(z,q)=[ijkl]
(107)
theta_i(w^',q)theta_j(x^',q)theta_k(y^'+pi/2,q)theta_l(z^'+pi/2,q)=ijkl,
(108)

gives a whopping 288 identities of the form

+/-[a_1a_2a_3a_4]+/-[b_1b_2b_3b_4]=+/-a_1^'a_2^'a_3^'a_4^'+/-b_1^'b_2^'b_3^'b_4^'.
(109)

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

xi=(theta_1(z))/(theta_4(z)),
(110)

and plug into (◇)

((dxi)/(dz))^2=(theta_2^2-xi^2theta_3^2)(theta_3^2-xi^2theta_2^2).
(111)

Now write

xi(theta_3)/(theta_2)=y
(112)

and

ztheta_3^2=u.
(113)

Then

((dy)/(du))^2=(1-y^2)(1-k^2y^2),
(114)

where the elliptic modulus is defined by

k=k(q)=(theta_2^2(q))/(theta_3^2(q)).
(115)

Define also the complementary elliptic modulus

k^'=k^'(q)=(theta_4^2(-q))/(theta_3^2(q)).
(116)

Now, since

theta_2^4+theta_4^4=theta_3^4,
(117)

we have shown

k^2+k^('2)=1.
(118)

The solution to the equation is

y=(theta_3)/(theta_2)(theta_1(utheta_3^(-2)|tau))/(theta_4(utheta_3^(-2)|tau))=sn(u,k),
(119)

which is a Jacobi elliptic function with periods

4K(k)=2pitheta_3^2(q)
(120)

and

2iK^'(k)=pitautheta_3^2(q).
(121)

Letting K(k) be the complete elliptic integral of the first kind with modulus k, then

theta_2^2(q)=(2kK(k))/pi
(122)
theta_3^2(q)=(2K(k))/pi
(123)
theta_4^2(q)=(2k^'K(k))/pi,
(124)

where k^'=sqrt(1-k^2) 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 r_2(n) giving the number of representations of n by two squares via

theta_3^2(q)=sum_(n=0)^(infty)r_2(n)q^n
(125)
theta_4^2(q)=sum_(n=0)^(infty)(-1)^nr_2(n)q^n
(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.

See also

Blecksmith-Brillhart-Gerst Theorem, Dedekind Eta Function, Elliptic Function, Half-Period Ratio, Jacobi Elliptic Functions, Jacobi's Imaginary Transformation, Jacobi Triple Product, Landen's Formula, Mock Theta Function, Modular Equation, Mordell Integral, Neville Theta Functions, Nome, Pentagonal Number Theorem, Poincaré-Fuchs-Klein Automorphic Function, Quintuple Product Identity, Ramanujan Theta Functions, Schröter's Formula, Sum of Squares Function, Weber Functions

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/, http://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime1/, http://functions.wolfram.com/EllipticFunctions/EllipticTheta2/, http://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime2/, http://functions.wolfram.com/EllipticFunctions/EllipticTheta3/, http://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime3/, http://functions.wolfram.com/EllipticFunctions/EllipticTheta4/, http://functions.wolfram.com/EllipticFunctions/EllipticThetaPrime4/

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References

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