Handbook of mathematics for engineers and scienteists part 144 pps

18.13. ELLIPTIC INTEGRALS 969 18.13. Elliptic Integrals 18.13.1. Complete Elliptic Integrals 18.13.1-1. Definitions. Properties. Conversion formulas. Complete elliptic integral of the first kind: K(k)=  π/2 0 dα √ 1 – k 2 sin 2 α =  1 0 dx  (1 – x 2 )(1 – k 2 x 2 ) . Complete elliptic integral of the second kind: E(k)=  π/2 0 √ 1 – k 2 sin 2 αdα=  1 0 √ 1 – k 2 x 2 √ 1 – x 2 dx. The argument k is called the elliptic modulus (k 2 < 1). Notation: k  = √ 1 – k 2 , K  (k)=K(k  ), E  (k)=E(k  ), where k  is the complementary modulus. Properties: K(–k)=K(k), E(–k)=E(k); K(k)=K  (k  ), E(k)=E  (k  ); E(k) K  (k)+E  (k) K(k)–K(k) K  (k)= π 2 . Conversion formulas for complete elliptic integrals: K  1 – k  1 + k   = 1 + k  2 K(k), E  1 – k  1 + k   = 1 1 + k   E(k)+k  K(k)  , K  2 √ k 1 + k  =(1 + k) K(k), E  2 √ k 1 + k  = 1 1 + k  2 E(k)–(k  ) 2 K(k)  . 18.13.1-2. Representation of complete elliptic integrals in series form. Representation of complete elliptic integrals in the form of series in powers of the modulus k: K(k)= π 2  1 +  1 2  2 k 2 +  1×3 2×4  2 k 4 + ···+  (2n – 1)!! (2n)!!  2 k 2n + ···  , E(k)= π 2  1 –  1 2  2 k 2 1 –  1×3 2×4  2 k 4 3 – ···–  (2n – 1)!! (2n)!!  2 k 2n 2n – 1 – ···  . 970 SPECIAL FUNCTIONS AND THEIR PROPERTIES Representation of complete elliptic integrals in the form of series in powers of the comple- mentary modulus k  = √ 1 – k 2 : K(k)= π 1 +k   1 +  1 2  2  1 –k  1 +k   2 +  1×3 2×4  2  1 –k  1 +k   4 + ···+  (2n –1)!! (2n)!!  2  1 –k  1 +k   2n + ···  , K(k)=ln 4 k  +  1 2  2  ln 4 k  – 2 1×2  (k  ) 2 +  1×3 2×4  2  ln 4 k  – 2 1×2 – 2 3×4  (k  ) 4 +  1×3×5 2×4×6  2  ln 4 k  – 2 1×2 – 2 3×4 – 2 5×6  (k  ) 6 + ···; E(k)= π(1 +k  ) 4  1 + 1 2 2 –  1 –k  1 +k   2 + 1 2 (2×4) 2  1 –k  1 +k   4 + ···+  (2n –3)!! (2n)!!  2  1 –k  1 +k   2n + ···  , E(k)=1 + 1 2  ln 4 k  – 1 1×2  (k  ) 2 + 1 2 ×3 2 2 ×4  ln 4 k  – 2 1×2 – 1 3×4  (k  ) 4 + 1 2 ×3 2 ×5 2 2 ×4 2 ×6  ln 4 k  – 2 1×2 – 2 3×4 – 1 5×6  (k  ) 6 + ···. 18.13.1-3. Differentiation formulas. Differential equations. Differentiation formulas: d K(k) dk = E(k) k(k  ) 2 – K(k) k , d E(k) dk = E(k)–K(k) k . The functions K(k)andK  (k) satisfy the second-order linear ordinary differential equa- tion d dk  k(1 – k 2 ) d K dk  – k K = 0. The functions E(k)andE  (k)–K  (k) satisfy the second-order linear ordinary differential equation (1 – k 2 ) d dk  k d E dk  + k E = 0. 18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals) 18.13.2-1. Definitions. Properties. Elliptic integral of the first kind: F (ϕ, k)=  ϕ 0 dα √ 1 – k 2 sin 2 α =  sin ϕ 0 dx  (1 – x 2 )(1 – k 2 x 2 ) . Elliptic integral of the second kind: E(ϕ, k)=  ϕ 0 √ 1 – k 2 sin 2 αdα=  sin ϕ 0 √ 1 – k 2 x 2 √ 1 – x 2 dx. 18.13. ELLIPTIC INTEGRALS 971 Elliptic integral of the third kind: Π(ϕ, n, k)=  ϕ 0 dα (1 – n sin 2 α) √ 1 – k 2 sin 2 α =  sin ϕ 0 dx (1 – nx 2 )  (1 – x 2 )(1 – k 2 x 2 ) . The quantity k is called the elliptic modulus (k 2 < 1), k  = √ 1 – k 2 is the complementary modulus,andn is the characteristic parameter. Complete elliptic integrals: K(k)=F  π 2 , k  , E(k)=E  π 2 , k  , K  (k)=F  π 2 , k   , E  (k)=E  π 2 , k   . Properties of elliptic integrals: F (–ϕ, k)=–F (ϕ, k), F (nπ ϕ, k)=2n K(k) F (ϕ, k); E(–ϕ, k)=–E(ϕ, k), E(nπ ϕ, k)=2n E(k) E(ϕ, k). 18.13.2-2. Conversion formulas. Conversion formulas for elliptic integrals (first set): F  ψ, 1 k  = kF(ϕ, k), E  ψ, 1 k  = 1 k  E(ϕ, k)–(k  ) 2 F (ϕ, k)  , where the angles ϕ and ψ are related by sin ψ = k sin ϕ,cosψ =  1 – k 2 sin 2 ϕ. Conversion formulas for elliptic integrals (second set): F  ψ, 1 – k  1 + k   =(1 + k  )F (ϕ, k), E  ψ, 1 – k  1 + k   = 2 1 + k   E(ϕ, k)+k  F (ϕ, k)  – 1 – k  1 + k  sin ψ, where the angles ϕ and ψ are related by tan(ψ – ϕ)=k  tan ϕ. Transformation formulas for elliptic integrals (third set): F  ψ, 2 √ k 1 + k  =(1 + k)F (ϕ, k), E  ψ, 2 √ k 1 + k  = 1 1 + k  2E(ϕ, k)–(k  ) 2 F (ϕ, k)+2k sin ϕ cos ϕ 1 + k sin 2 ϕ  1 – k 2 sin 2 ϕ  , where the angles ϕ and ψ are related by sin ψ = (1 + k)sinϕ 1 + k sin 2 ϕ . 972 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.13.2-3. Trigonometric expansions. Trigonometric expansions for small k and ϕ: F (ϕ, k)= 2 π K(k)ϕ –sinϕ cos ϕ  a 0 + 2 3 a 1 sin 2 ϕ + 2×4 3×5 a 2 sin 4 ϕ + ···  , a 0 = 2 π K(k)–1, a n = a n–1 –  (2n – 1)!! (2n)!!  2 k 2n ; E(ϕ, k)= 2 π E(k)ϕ –sinϕ cos ϕ  b 0 + 2 3 b 1 sin 2 ϕ + 2×4 3×5 b 2 sin 4 ϕ + ···  , b 0 = 1 – 2 π E(k), b n = b n–1 –  (2n – 1)!! (2n)!!  2 k 2n 2n – 1 . Trigonometric expansions for k → 1: F (ϕ, k)= 2 π K  (k)lntan  ϕ 2 + π 4  – tan ϕ cos ϕ  a  0 – 2 3 a  1 tan 2 ϕ + 2×4 3×5 a  2 tan 4 ϕ – ···  , a  0 = 2 π K  (k)–1, a  n = a  n–1 –  (2n – 1)!! (2n)!!  2 (k  ) 2n ; E(ϕ, k)= 2 π E  (k)lntan  ϕ 2 + π 4  + tan ϕ cos ϕ  b  0 – 2 3 b  1 tan 2 ϕ + 2×4 3×5 b  2 tan 4 ϕ – ···  , b  0 = 2 π E  (k)–1, b  n = b  n–1 –  (2n – 1)!! (2n)!!  2 (k  ) 2n 2n – 1 . 18.14. Elliptic Functions An elliptic function is a function that is the inverse of an elliptic integral. An elliptic function is a doubly periodic meromorphic function of a complex variable. All its periods can be written in the form 2mω 1 + 2nω 2 with integer m and n,whereω 1 and ω 2 are a pair of (primitive) half-periods. The ratio τ = ω 2 /ω 1 is a complex quantity that may be considered to have a positive imaginary part, Im τ > 0. Throughout the rest of this section, the following brief notation will be used: K = K(k) and K  = K(k  ) are complete elliptic integrals with k  = √ 1 – k 2 . 18.14.1. Jacobi Elliptic Functions 18.14.1-1. Definitions. Simple properties. Special cases. When the upper limit ϕ of the incomplete elliptic integral of the first kind u =  ϕ 0 dα √ 1 – k 2 sin 2 α = F (ϕ, k) is treated as a function of u, the following notation is used: u =amϕ. 18.14. ELLIPTIC FUNCTIONS 973 Naming: ϕ is the amplitude and u is the argument. Jacobi elliptic functions: sn u =sinϕ =sinamu (sine amplitude), cn u =cosϕ =cosamu (cosine amplitude), dn u =  1 – k 2 sin 2 ϕ = dϕ du (delta amlplitude). Along with the brief notations sn u,cnu,dnu, the respective full notations are also used: sn(u, k), cn(u, k), dn(u, k). Simple properties: sn(–u)=–snu,cn(–u)=cnu,dn(–u)=dnu; sn 2 u +cn 2 u = 1, k 2 sn 2 u +dn 2 u = 1,dn 2 u – k 2 cn 2 u = 1 – k 2 , where i 2 =–1. Jacobi functions for special values of the modulus (k = 0 and k = 1): sn(u, 0)=sinu,cn(u, 0)=cosu,dn(u, 0)=1; sn(u, 1)=tanhu,cn(u, 1)= 1 cosh u ,dn(u, 1)= 1 cosh u . Jacobi functions for special values of the argument: sn( 1 2 K, k)= 1 √ 1 + k  ,cn( 1 2 K, k)=  k  1 + k  ,dn( 1 2 K, k)= √ k  ; sn(K, k)=1,cn(K, k)=0,dn(K, k)=k  . 18.14.1-2. Reduction formulas. sn(u K)= cn u dn u ,cn(u K)= k  sn u dn u , dn(u K)= k  dn u ; sn(u 2 K)=–snu,cn(u 2 K)=–cnu, dn(u 2 K)=dnu; sn(u + i K  )= 1 k sn u ,cn(u + i K  )=– i k dn u sn u , dn(u + i K  )=–i cn u sn u ; sn(u + 2i K  )=snu,cn(u + 2i K  )=–cnu, dn(u + 2i K  )=–dnu; sn(u + K +i K  )= dn u k cn u ,cn(u + K+i K  )=–i k  k cn u , dn(u + K+i K  )=ik  sn u cn u ; sn(u + 2 K +2i K  )=–snu,cn(u + 2 K +2i K  )=cnu, dn(u + 2 K +2i K  )=–dnu. 18.14.1-3. Periods, zeros, poses, and residues. TABLE 18.4 Periods, zeros, poles, and residues of the Jacobian elliptic functions ( m, n = 0, 1, 2, ; i 2 =–1) Functions Periods Zeros Poles Residues sn u 4m K +2n K  i 2m K +2n K  i 2m K +(2n +1) K  i (–1) m 1 k cn u (4m+2n) K+2n K  i (2m+1) K+2n K  i 2m K +(2n +1) K  i (–1) m–1 i k dn u 2m K +4n K  i (2m+1) K+(2n+1) K  i 2m K +(2n +1) K  i (–1) n–1 i 974 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.14.1-4. Double-argument formulas. sn(2u)= 2 sn u cn u dn u 1 – k 2 sn 4 u = 2 sn u cn u dn u cn 2 u +sn 2 u dn 2 u , cn(2u)= cn 2 u –sn 2 u dn 2 u 1 – k 2 sn 4 u = cn 2 u –sn 2 u dn 2 u cn 2 u +sn 2 u dn 2 u , dn(2u)= dn 2 u – k 2 sn 2 u cn 2 u 1 – k 2 sn 4 u = dn 2 u +cn 2 u (dn 2 u – 1) dn 2 u –cn 2 u (dn 2 u – 1) . 18.14.1-5. Half-argument formulas. sn 2 u 2 = 1 k 2 1 –dnu 1 +cnu = 1 –cnu 1 +dnu , cn 2 u 2 = cn u +dnu 1 +dnu = 1 – k 2 k 2 1 –dnu dn u –cnu , dn 2 u 2 = cn u +dnu 1 +cnu =(1 – k 2 ) 1 –cnu dn u –cnu . 18.14.1-6. Argument addition formulas. sn(u v)= sn u cn v dn v sn v cn u dn u 1 – k 2 sn 2 u sn 2 v , cn(u v)= cn u cn v sn u sn v dn u dn v 1 – k 2 sn 2 u sn 2 v , dn(u v)= dn u dn v k 2 sn u sn v cn u cn v 1 – k 2 sn 2 u sn 2 v . 18.14.1-7. Conversion formulas. Table 18.5 presents conversion formulas for Jacobi elliptic functions. If k > 1,then k 1 = 1/k < 1. Elliptic functions with real modulus can be reduced, using the first set of conversion formulas, to elliptic functions with a modulus lying between 0 and 1. 18.14.1-8. Descending Landen transformation (Gauss’s transformation). Notation: μ =     1 – k  1 + k      , v = u 1 + μ . Descending transformations: sn(u, k)= (1 + μ)sn(v, μ 2 ) 1 + μ sn 2 (v, μ 2 ) ,cn(u, k)= cn(v, μ 2 )dn(v, μ 2 ) 1 + μ sn 2 (v, μ 2 ) ,dn(u, k)= dn 2 (v, μ 2 )+μ – 1 1 + μ –dn 2 (v, μ 2 ) . 18.14. ELLIPTIC FUNCTIONS 975 TABLE 18.5 Conversion formulas for Jacobi elliptic functions. Full notation is used: sn(u, k), cn(u, k), dn(u, k) u 1 k 1 sn(u 1 , k 1 ) cn(u 1 , k 1 ) dn(u 1 , k 1 ) ku 1 k k sn(u, k) dn(u, k) cn(u, k) iu k  i sn(u, k) cn(u, k) 1 cn(u, k) dn(u, k) cn(u, k) k  u i k k  k  sn(u, k) dn(u, k) cn(u, k) dn(u, k) 1 dn(u, k) iku i k  k ik sn(u, k) dn(u, k) 1 dn(u, k) cn(u, k) dn(u, k) ik  u 1 k  ik  sn(u, k) cn(u, k) dn(u, k) cn(u, k) 1 cn(u, k) (1 +k)u 2 √ k 1 +k (1 +k)sn(u, k) 1 +k sn 2 (u, k) cn(u, k)dn(u, k) 1 +k sn 2 (u, k) 1 –k sn 2 (u, k) 1 +k sn 2 (u, k) (1 +k  )u 1 –k  1 +k  (1 +k  )sn(u, k)cn(u, k) dn(u, k) 1 –(1 +k  )sn 2 (u, k) dn(u, k) 1 –(1 –k  )sn 2 (u, k) dn(u, k) 18.14.1-9. Ascending Landen transformation. Notation: μ = 4k (1 + k) 2 , σ =    1 – k 1 + k    , v = u 1 + σ . Ascending transformations: sn(u, k)=(1 +σ) sn(v, μ)cn(v, μ) dn(v, μ) ,cn(u, k)= 1 +σ μ dn 2 (v, μ)–σ dn(v, μ) ,dn(u, k)= 1 –σ μ dn 2 (v, μ)+σ dn(v, μ) . 18.14.1-10. Series representation. Representation Jacobi functions in the form of power series in u: sn u = u – 1 3! (1 + k 2 )u 3 + 1 5! (1 + 14k 2 + k 4 )u 5 – 1 7! (1 + 135k 2 + 135k 4 + k 6 )u 7 + ··· , cn u = 1 – 1 2! u 2 + 1 4! (1 + 4k 2 )u 4 – 1 6! (1 + 44k 2 + 16k 4 )u 6 + ··· , dn u = 1 – 1 2! k 2 u 2 + 1 4! k 2 (4 + k 2 )u 4 – 1 6! k 2 (16 + 44k 2 + k 4 )u 6 + ··· , am u = u – 1 3! k 2 u 3 + 1 5! k 2 (4 + k 2 )u 5 – 1 7! k 2 (16 + 44k 2 + k 4 )u 7 + ··· . These functions converge for |u| < |K(k  )|. Representation Jacobi functions in the form of trigonometric series: sn u = 2π k K √ q ∞  n=1 q n 1 – q 2n–1 sin  (2n – 1) πu 2 K  , . E(k)–(k  ) 2 K(k)  . 18.13.1-2. Representation of complete elliptic integrals in series form. Representation of complete elliptic integrals in the form of series in powers of the modulus k: K(k)= π 2  1 +  1 2  2 k 2 +  1×3 2×4  2 k 4 +. 1)!! (2n)!!  2 k 2n 2n – 1 – ···  . 970 SPECIAL FUNCTIONS AND THEIR PROPERTIES Representation of complete elliptic integrals in the form of series in powers of the comple- mentary modulus k  = √ 1 – k 2 : K(k)= π 1. functions for special values of the modulus (k = 0 and k = 1): sn(u, 0)=sinu,cn(u, 0)=cosu,dn(u, 0)=1; sn(u, 1)=tanhu,cn(u, 1)= 1 cosh u ,dn(u, 1)= 1 cosh u . Jacobi functions for special values of