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Jacobi Theta Functions  
  
1999   02:03 مساءً   date: 23-11-2018
Author : Abramowitz, M. and Stegun, I. A.
Book or Source : Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
Page and Part : ...


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Date: 22-11-2018 433
Date: 28-11-2018 512
Date: 27-11-2018 1067

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 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[nzq], and their derivatives are given by EllipticThetaPrime[nzq].

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_i theta_i(z+pi)/theta_i(z) theta_i(z+taupi)/theta_i(z)
theta_1 -1 -N
theta_2 -1 N
theta_3 1 N
theta_4 1 -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  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

(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

(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

=

(50)

=

(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

(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

= 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)

= -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)

= 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)

= 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_0x=Q_1y=Q_2, and z=Q_3.

theta function Sloane Eulerian Jacobian
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)) A089799 2q^(1/8)((2)^2)/((1)) 2q^(1/8)wxz
theta_2(q) A089800 2q^(1/4)((4)^2)/((2)) 2q^(1/4)wx^2
1/2[theta_3(q^(1/3))-theta_3(q^3)] A089801 q^(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))] A089802 q^(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))] A089805 q^(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))] A080995 q^(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)] A089806 q^(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))] A089812 q^(1/8)((1)^2(6))/((2)(3)) q^(1/8)wy^2|xz|q^3
1/2[theta_2(q)-3theta_2(q^9)] A089813 q^(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 wxy, and z and , and , related by

= -w+x+y+z

(103)

= w-x-y+z

(104)

= w+x-y+z

(105)

= 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)

(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

 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

(116)

Now, since

 theta_2^4+theta_4^4=theta_3^4,

(117)

we have shown

(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

(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) =

(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 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.


 

REFERENCES:

 

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 576-579, 1972.

Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961.

Berndt, B. C. "Theta-Functions and Modular Equations." Ch. 25 in Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 138-244, 1994.

Borwein, J. M. and Borwein, P. B. "Theta Functions and the Arithmetic-Geometric Mean Iteration." Ch. 2 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 33-61, 1987.

Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.

Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.

Euler, L. Opera Omnia, Series Prima, Vol. 20. Leipzig, Germany, 1912.

Golden, S. Introduction to Theoretical Physical Chemistry. New York: Addison-Wesley, 1961.

Hermite, C. Oeuvres Mathématiques. Paris, 1905-1917.

Jacobi, C. G. J. "Suite des notices sur les fonctions elliptiques." J. reine angew. Math. 3, 303-310, 1828. Reprinted in Gesammelte Mathematische Werke, Vol. 1, pp. 255-263.

Jacobi, C. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, 1829. Reprinted in Gesammelte Mathematische Werke, Band. 1. Providence, RI: Amer. Math. Soc., pp. 97-239, 1969.

Klein, F. Vorlesungen über die Theorie der elliptischen Modulfunctionen, 2 vols. Leipzig, Germany: Teubner, 1890-92.

Kronecker, L. "Bemerkungen über die Jacobi'schen Thetaformeln." J. reine angew. Math. 102, 260-272, 1887.

Levine, I. A. Physical Chemistry, 5th ed. New York: McGraw-Hill, 2002.

Melzak, Z. A. Companion to Concrete Mathematics, Vol. 1. New York: Wiley, 1973.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 430-432, 1953.

Oberhettinger, F. and Magnus, W. Anwendung der Elliptischen Funktionen in Physik und Technik. Berlin: Springer-Verlag, 1949.

Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.

Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972. Also reprinted in English translation in Edwards, H. M. Appendix. Riemann's Zeta Function. New York: Dover, pp. 299-305, 2001.

Scheibner, W. "Über die Producte von drei und vier Thetafunctionen." J. reine angew. Math. 102, 255-259, 1887.

Sloane, N. J. A. Sequences A000122, A002448, A022597, A089798, A089799, A089800, A089801, A089802, A089803,A089804, A089805, A080995, A089806, A089807, A089810, A089811, A089812, and A089813 in "The On-Line Encyclopedia of Integer Sequences."

Tannery, J. and Molk, J. Elements de la Theorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars, 1893-1902.

Tölke, F. "Theta-Funktionen" and "Logarithmen der Theta-Funktionen." Chs. 1-2 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 1-83, 1966.

Tölke, F. Praktische Funktionenlehre, fünfter Band: Allgemeine Weierstraßsche Funktionen und Ableitungen nach dem Parameter. Integrale der Theta-Funktionen und Bilinear-Entwicklungen. Berlin: Springer-Verlag, 1968.

Trott, M. "The Mathematica Guidebooks Additional Material: Theta Function Identities." http://www.mathematicaguidebooks.org/additions.shtml#S_3_08.

Weber, H. Elliptische Funktionen und algebraische Zahlen. Brunswick, Germany, 1891.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Zucker, J. "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.




الجبر أحد الفروع الرئيسية في الرياضيات، حيث إن التمكن من الرياضيات يعتمد على الفهم السليم للجبر. ويستخدم المهندسون والعلماء الجبر يومياً، وتعول المشاريع التجارية والصناعية على الجبر لحل الكثير من المعضلات التي تتعرض لها. ونظراً لأهمية الجبر في الحياة العصرية فإنه يدرّس في المدارس والجامعات في جميع أنحاء العالم. ويُعجب الكثير من الدارسين للجبر بقدرته وفائدته الكبيرتين، إذ باستخدام الجبر يمكن للمرء أن يحل كثيرًا من المسائل التي يتعذر حلها باستخدام الحساب فقط.وجاء اسمه من كتاب عالم الرياضيات والفلك والرحالة محمد بن موسى الخورازمي.


يعتبر علم المثلثات Trigonometry علماً عربياً ، فرياضيو العرب فضلوا علم المثلثات عن علم الفلك كأنهما علمين متداخلين ، ونظموه تنظيماً فيه لكثير من الدقة ، وقد كان اليونان يستعملون وتر CORDE ضعف القوسي قياس الزوايا ، فاستعاض رياضيو العرب عن الوتر بالجيب SINUS فأنت هذه الاستعاضة إلى تسهيل كثير من الاعمال الرياضية.

تعتبر المعادلات التفاضلية خير وسيلة لوصف معظم المـسائل الهندسـية والرياضـية والعلمية على حد سواء، إذ يتضح ذلك جليا في وصف عمليات انتقال الحرارة، جريان الموائـع، الحركة الموجية، الدوائر الإلكترونية فضلاً عن استخدامها في مسائل الهياكل الإنشائية والوصف الرياضي للتفاعلات الكيميائية.
ففي في الرياضيات, يطلق اسم المعادلات التفاضلية على المعادلات التي تحوي مشتقات و تفاضلات لبعض الدوال الرياضية و تظهر فيها بشكل متغيرات المعادلة . و يكون الهدف من حل هذه المعادلات هو إيجاد هذه الدوال الرياضية التي تحقق مشتقات هذه المعادلات.