976 SPECIAL FUNCTIONS AND THEIR PROPERTIES cn u = 2π k K √ q ∞  n=1 q n 1 + q 2n–1 cos  (2n – 1) πu 2 K  , dn u = π 2 K + 2π K ∞  n=1 q n 1 + q 2n cos  nπu K  , am u = πu 2 K + 2 ∞  n=1 1 n q n 1 + q 2n sin  nπu K  , where q =exp(–π K  / K), K = K(k), K  = K(k  ), and k  = √ 1 – k 2 . 18.14.1-11. Derivatives and integrals. Derivatives: d du sn u =cnu dn u, d du cn u =–snu dn u, d du dn u =–k 2 sn u cn u. Integrals:  sn udu= 1 k ln(dn u – k cn u)=– 1 k ln(dn u + k cn u),  cn udu= 1 k arccos(dn u)= 1 k arcsin(k sn u),  dn udu= arcsin(sn u)=amu. The arbitrary additive constant C in the integrals is omitted. 18.14.2. Weierstrass Elliptic Function 18.14.2-1. Infinite series representation. Some properties. The Weierstrass elliptic function (or Weierstrass ℘-function) is defined as ℘(z)=℘(z|ω 1 , ω 2 )= 1 z 2 +  m,n  1 (z – 2mω 1 – 2nω 2 ) 2 – 1 (2mω 1 + 2nω 2 ) 2  , where the summation is assumed over all integer m and n, except for m = n = 0.This function is a complex, double periodic function of a complex variable z with periods 2ω 1 and 2ω 1 : ℘(–z)=℘(z), ℘(z + 2mω 1 + 2nω 2 )=℘(z), where m, n = 0, 1, 2, and Im(ω 2 /ω 1 ) ≠ 0. The series defining the Weierstrass ℘- function converges everywhere except for second-order poles located at z mn =2mω 1 +2nω 2 . Argument addition formula: ℘(z 1 + z 2 )=–℘(z 1 )–℘(z 2 )+ 1 4  ℘  (z 1 )–℘  (z 2 ) ℘(z 1 )–℘(z 2 )  2 . 18.14. ELLIPTIC FUNCTIONS 977 18.14.2-2. Representation in the form of a definite integral. The Weierstrass function ℘ = ℘(z, g 2 , g 3 )=℘(z|ω 1 , ω 2 )isdefined implicitly by the elliptic integral: z =  ∞ ℘ dt  4t 3 – g 2 t – g 3 =  ∞ ℘ dt 2 √ (t – e 1 )(t – e 2 )(t – e 3 ) . The parameters g 2 and g 3 are known as the invariants. The parameters e 1 , e 2 , e 3 , which are the roots of the cubic equation 4z 3 – g 2 z – g 3 = 0, are related to the half-periods ω 1 , ω 2 and invariants g 2 , g 3 by e 1 = ℘(ω 1 ), e 2 = ℘(ω 1 + ω 2 ), e 1 = ℘(ω 2 ), e 1 + e 2 + e 3 = 0, e 1 e 2 + e 1 e 3 + e 2 e 3 =– 1 4 g 2 , e 1 e 2 e 3 = 1 4 g 3 . Homogeneity property: ℘(z, g 2 , g 3 )=λ 2 ℘(λz, λ –4 g 2 , λ –6 g 3 ). 18.14.2-3. Representation as a Laurent series. Differential equations. The Weierstrass ℘-function can be expanded into a Laurent series: ℘(z)= 1 z 2 + g 2 20 z 2 + g 3 28 z 4 + g 2 2 1200 z 6 + 3g 2 g 3 6160 z 8 + ··· = 1 z 2 + ∞  k=2 a k z 2k–2 , a k = 3 (k – 3)(2k + 1) k–2  m=2 a m a k–m for k ≥ 4, 0 < |z| <min(|ω 1 |, |ω 2 |). The Weierstrass ℘-function satisfies the first-order and second-order nonlinear differen- tial equations: (℘  z ) 2 = 4℘ 3 – g 2 ℘ – g 3 , ℘  zz = 6℘ 2 – 1 2 g 2 . 18.14.2-4. Connection with Jacobi elliptic functions. Direct and inverse representations of the Weierstrass elliptic function via Jacobi elliptic functions: ℘(z)=e 1 +(e 1 – e 3 ) cn 2 w sn 2 w = e 2 +(e 1 – e 3 ) dn 2 w sn 2 w = e 3 + e 1 – e 3 sn 2 w ; sn w =  e 1 – e 3 ℘(z)–e 3 ,cnw =  ℘(z)–e 1 ℘(z)–e 3 ,dnw =  ℘(z)–e 2 ℘(z)–e 3 ; w = z √ e 1 – e 3 = K z/ω 1 . The parameters are related by k =  e 2 – e 3 e 1 – e 3 , k  =  e 1 – e 2 e 1 – e 3 , K = ω 1 √ e 1 – e 3 , i K  = ω 2 √ e 1 – e 3 . 978 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.15. Jacobi Theta Functions 18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties 18.15.1-1. Definition of the Jacobi theta functions. The Jacobi theta functions are defined by the following series: ϑ 1 (v)=ϑ 1 (v, q)=ϑ 1 (v|τ)=2 ∞  n=0 (–1) n q (n+1/2) 2 sin[(2n + 1)πv]=i ∞  n=–∞ (–1) n q (n–1/2) 2 e iπ(2n–1)v , ϑ 2 (v)=ϑ 2 (v, q)=ϑ 2 (v|τ)=2 ∞  n=0 q (n+1/2) 2 cos[(2n + 1)πv]= ∞  n=–∞ q (n–1/2) 2 e iπ(2n–1)v , ϑ 3 (v)=ϑ 3 (v, q)=ϑ 3 (v|τ)=1 + 2 ∞  n=0 q n 2 cos(2nπv)= ∞  n=–∞ q n 2 e 2iπnv , ϑ 4 (v)=ϑ 4 (v, q)=ϑ 4 (v|τ)=1 + 2 ∞  n=0 (–1) n q n 2 cos(2nπv)= ∞  n=–∞ (–1) n q n 2 e 2iπnv , where v is a complex variable and q = e iπτ is a complex parameter (τ has a positive imaginary part). 18.15.1-2. Simplest properties. The Jacobi theta functions are periodic entire functions that possess the following properties: ϑ 1 (v) odd, has period 2, vanishes at v = m + nτ; ϑ 2 (v) even, has period 2, vanishes at v = m + nτ + 1 2 ; ϑ 3 (v) even, has period 1, vanishes at v = m +(n + 1 2 )τ + 1 2 ; ϑ 4 (v) even, has period 1, vanishes at v = m +(n + 1 2 )τ. Here, m, n = 0, 1, 2, Remark. The theta functions are not elliptic functions. The very good convergence of their series allows the computation of various elliptic integrals and elliptic functions using the relations given above in Paragraph 18.15.1-1. 18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 18.15.2-1. Linear and quadratic relations. Linear relations (first set): ϑ 1  v + 1 2  = ϑ 2 (v), ϑ 2  v + 1 2  =–ϑ 1 (v), ϑ 3  v + 1 2  = ϑ 4 (v), ϑ 4  v + 1 2  = ϑ 3 (v), ϑ 1  v + τ 2  = ie –iπ  v+ τ 4  ϑ 4 (v), ϑ 2  v + τ 2  = e –iπ  v+ τ 4  ϑ 3 (v), ϑ 3  v + τ 2  = e –iπ  v+ τ 4  ϑ 2 (v), ϑ 4  v + τ 2  = ie –iπ  v+ τ 4  ϑ 1 (v). 18.15. JACOBI THETA FUNCTIONS 979 Linear relations (second set): ϑ 1 (v|τ + 1)=e iπ/4 ϑ 1 (v|τ), ϑ 2 (v|τ + 1)=e iπ/4 ϑ 2 (v|τ), ϑ 3 (v|τ + 1)=ϑ 4 (v|τ), ϑ 4 (v|τ + 1)=ϑ 3 (v|τ), ϑ 1  v τ    – 1 τ  = 1 i  τ i e iπv 2 /τ ϑ 1 (v|τ), ϑ 2  v τ    – 1 τ  =  τ i e iπv 2 /τ ϑ 4 (v|τ), ϑ 3  v τ    – 1 τ  =  τ i e iπv 2 /τ ϑ 3 (v|τ), ϑ 4  v τ    – 1 τ  =  τ i e iπv 2 /τ ϑ 2 (v|τ). Quadratic relations: ϑ 2 1 (v)ϑ 2 2 (0)=ϑ 2 4 (v)ϑ 2 3 (0)–ϑ 2 3 (v)ϑ 2 4 (0), ϑ 2 1 (v)ϑ 2 3 (0)=ϑ 2 4 (v)ϑ 2 2 (0)–ϑ 2 2 (v)ϑ 2 4 (0), ϑ 2 1 (v)ϑ 2 4 (0)=ϑ 2 3 (v)ϑ 2 2 (0)–ϑ 2 2 (v)ϑ 2 3 (0), ϑ 2 4 (v)ϑ 2 4 (0)=ϑ 2 3 (v)ϑ 2 3 (0)–ϑ 2 2 (v)ϑ 2 2 (0). 18.15.2-2. Representation of the theta functions in the form of infinite products. ϑ 1 (v)=2q 0 q 1/4 sin(πv) ∞  n=1  1 – 2q 2n cos(2πv)+q 4n  , ϑ 2 (v)=2q 0 q 1/4 cos(πv) ∞  n=1  1 + 2q 2n cos(2πv)+q 4n  , ϑ 3 (v)=q 0 ∞  n=1  1 + 2q 2n–1 cos(2πv)+q 4n–2  , ϑ 4 (v)=q 0 ∞  n=1  1 – 2q 2n–1 cos(2πv)+q 4n–2  , where q 0 = ∞  n=1 (1 – q 2n ). 18.15.2-3. Connection with Jacobi elliptic functions. Representations of Jacobi elliptic functions in terms of the theta functions: sn w = ϑ 3 (0) ϑ 2 (0) ϑ 1 (v) ϑ 4 (v) ,cnw = ϑ 4 (0) ϑ 2 (0) ϑ 2 (v) ϑ 4 (v) ,dnw = ϑ 4 (0) ϑ 3 (0) ϑ 3 (v) ϑ 4 (v) , w = 2 K v. The parameters are related by k = ϑ 2 2 (0) ϑ 2 3 (0) , k  = ϑ 2 4 (0) ϑ 2 3 (0) , K = π 2 ϑ 2 3 (0), K  =–iτ K. 980 SPECIAL FUNCTIONS AND THEIR PROPERTIES TABLE 18.6 The Mathieu functions ce n =ce n (x, q)andse n =se n (x, q) (for odd n, functions ce n and se n are 2π-periodic, and for even n,theyareπ-periodic); definite eigenvalues a = a n (q)anda = b n (q) correspond to each value of parameter q Mathieu functions Recurrence relations for coefficients Normalization conditions ce 2n = ∞  m=0 A 2n 2m cos 2mx qA 2n 2 = a 2n A 2n 0 ; qA 2n 4 =(a 2n –4)A 2n 2 –2qA 2n 0 ; qA 2n 2m+2 =(a 2n –4m 2 )A 2n 2m –qA 2n 2m–2 , m ≥ 2 (A 2n 0 ) 2 + ∞  m=0 (A 2n 2m ) 2 =  2 if n = 0 1 if n ≥ 1 ce 2n+1 = ∞  m=0 A 2n+1 2m+1 cos(2m+1)x qA 2n+1 3 =(a 2n+1 –1–q)A 2n+1 1 ; qA 2n+1 2m+3 =[a 2n+1 –(2m+1) 2 ]A 2n+1 2m+1 –qA 2n+1 2m–1 , m ≥ 1 ∞  m=0 (A 2n+1 2m+1 ) 2 = 1 se 2n = ∞  m=0 B 2n 2m sin 2mx, se 0 = 0 qB 2n 4 =(b 2n –4)B 2n 2 ; qB 2n 2m+2 =(b 2n –4m 2 )B 2n 2m –qB 2n 2m–2 , m ≥ 2 ∞  m=0 (B 2n 2m ) 2 = 1 se 2n+1 = ∞  m=0 B 2n+1 2m+1 sin(2m+1)x qB 2n+1 3 =(b 2n+1 –1–q)B 2n+1 1 ; qB 2n+1 2m+3 =[b 2n+1 –(2m+1) 2 ]B 2n+1 2m+1 –qB 2n+1 2m–1 , m ≥ 1 ∞  m=0 (B 2n+1 2m+1 ) 2 = 1 18.16. Mathieu Functions and Modified Mathieu Functions 18.16.1. Mathieu Functions 18.16.1-1. Mathieu equation and Mathieu functions. The Mathieu functions ce n (x, q)andse n (x, q) are periodical solutions of the Mathieu equation y  xx +(a – 2q cos 2x)y = 0. Such solutions exist for definite values of parameters a and q (those values of a are referred to as eigenvalues). The Mathieu functions are listed in Table 18.6. 18.16.1-2. Properties of the Mathieu functions. The Mathieu functions possess the following properties: ce 2n (x,–q)=(–1) n ce 2n  π 2 –x, q  ,ce 2n+1 (x,–q)=(–1) n se 2n+1  π 2 –x, q  , se 2n (x,–q)=(–1) n–1 se 2n  π 2 –x, q  ,se 2n+1 (x,–q)=(–1) n ce 2n+1  π 2 –x, q  . Selecting sufficiently large number m and omitting the term with the maximum number in the recurrence relations (indicated in Table 18.6), we can obtain approximate relations for eigenvalues a n (or b n ) with respect to parameter q. Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients A n m (or B n m )to zero, we obtain an algebraic equation for finding a n (q)(orb n (q)). 18.16. MAT H I E U FUNCTIONS AND MODIFIED MAT H I EU FUNCTIONS 981 For fixed real q ≠ 0, eigenvalues a n and b n are all real and different, while if q > 0 then a 0 < b 1 < a 1 < b 2 < a 2 < ···; if q < 0 then a 0 < a 1 < b 1 < b 2 < a 2 < a 3 < b 3 < b 4 < ··· . The eigenvalues possess the properties a 2n (–q)=a 2n (q), b 2n (–q)=b 2n (q), a 2n+1 (–q)=b 2n+1 (q). Tables of the eigenvalues a n = a n (q)andb n = b n (q) can be found in Abramowitz and Stegun (1964, chap. 20). The solution of the Mathieu equation corresponding to eigenvalue a n (or b n )hasn zeros on the interval 0 ≤ x < π (q is a real number). 18.16.1-3. Asymptotic expansions as q → 0 and q →∞. Listed below are two leading terms of asymptotic expansions of the Mathieu functions ce n (x, q)andse n (x, q), as well as of the corresponding eigenvalues a n (q)andb n (q), as q → 0: ce 0 (x, q)= 1 √ 2  1 – q 2 cos 2x  , a 0 (q)=– q 2 2 + 7q 4 128 ; ce 1 (x, q)=cosx – q 8 cos 3x, a 1 (q)=1 + q; ce 2 (x, q)=cos2x + q 4  1 – cos 4x 3  , a 2 (q)=4 + 5q 2 12 ; ce n (x, q)=cosnx + q 4  cos(n + 2)x n + 1 – cos(n – 2)x n – 1  , a n (q)=n 2 + q 2 2(n 2 – 1) (n ≥ 3); se 1 (x, q)=sinx – q 8 sin 3x, b 1 (q)=1 – q; se 2 (x, q)=sin2x – q sin 4x 12 , b 2 (q)=4 – q 2 12 ; se n (x, q)=sinnx – q 4  sin(n + 2)x n + 1 – sin(n – 2)x n – 1  , b n (q)=n 2 + q 2 2(n 2 – 1) (n ≥ 3). Asymptotic results as q →∞(–π/2 < x < π/2): a n (q) ≈ –2q + 2(2n + 1) √ q + 1 4 (2n 2 + 2n + 1), b n+1 (q) ≈ –2q + 2(2n + 1) √ q + 1 4 (2n 2 + 2n + 1), ce n (x, q) ≈ λ n q –1/4 cos –n–1 x  cos 2n+1 ξ exp(2 √ q sin x)+sin 2n+1 ξ exp(–2 √ q sin x)  , se n+1 (x, q) ≈ μ n+1 q –1/4 cos –n–1 x  cos 2n+1 ξ exp(2 √ q sin x)–sin 2n+1 ξ exp(–2 √ q sin x)  , where λ n and μ n are some constants independent of the parameter q,and ξ = 1 2 x + π 4 . 982 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.16.2. Modified Mathieu Functions The modified Mathieu functions Ce n (x, q)andSe n (x, q) are solutions of the modified Mathieu equation y  xx –(a – 2q cosh 2x)y = 0, with a = a n (q)anda = b n (q) being the eigenvalues of the Mathieu equation (see Subsection 18.16.1). The modified Mathieu functions are defined as Ce 2n+p (x, q)=ce 2n+p (ix, q)= ∞  k=0 A 2n+p 2k+p cosh[(2k + p)x], Se 2n+p (x, q)=–i se 2n+p (ix, q)= ∞  k=0 B 2n+p 2k+p sinh[(2k + p)x], where p may be equal to 0 and 1, and coefficients A 2n+p 2k+p and B 2n+p 2k+p are indicated in Subsection 18.16.1. 18.17. Orthogonal Polynomials All zeros of each of the orthogonal polynomials P n (x) considered in this section are real and simple. The zeros of the polynomials P n (x)andP n+1 (x) are alternating. For Legendre polynomials see Subsection 18.11.1. 18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials 18.17.1-1. Laguerre polynomials. The Laguerre polynomials L n = L n (x) satisfy the second-order linear ordinary differential equation xy  xx +(1 – x)y  x + ny = 0 and are defined by the formulas L n (x)= 1 n! e x d n dx n  x n e –x  = (–1) n n!  x n – n 2 x n–1 + n 2 (n – 1) 2 2! x n–2 + ···  . The first four polynomials have the form L 0 (x)=1, L 1 (x)=–x + 1, L 2 (x)= 1 2 (x 2 – 4x + 2), L 3 (x)= 1 6 (–x 3 + 9x 2 – 18x + 6). To calculate L n (x)forn ≥ 2, one can use the recurrence formulas L n+1 (x)= 1 n + 1  (2n + 1 – x)L n (x)–nL n–1 (x)  . The functions L n (x) form an orthonormal system on the interval 0 < x < ∞ with weight e –x :  ∞ 0 e –x L n (x)L m (x) dx =  0 if n ≠ m, 1 if n = m. The generating function is 1 1 – s exp  – sx 1 – s  = ∞  n=0 L n (x)s n , |s| < 1. . K. 980 SPECIAL FUNCTIONS AND THEIR PROPERTIES TABLE 18.6 The Mathieu functions ce n =ce n (x, q)andse n =se n (x, q) (for odd n, functions ce n and se n are 2π-periodic, and for even n,theyareπ-periodic);. as q → 0 and q →∞. Listed below are two leading terms of asymptotic expansions of the Mathieu functions ce n (x, q)andse n (x, q), as well as of the corresponding eigenvalues a n (q)andb n (q),. solutions exist for definite values of parameters a and q (those values of a are referred to as eigenvalues). The Mathieu functions are listed in Table 18.6. 18.16.1-2. Properties of the Mathieu