Incomplete Elliptic Integral of the First Kind ELLIPTIC FUNCTIONS
35 Incomplete Elliptic Integral of the First Kind ELLIPTIC FUNCTIONS
35.1. u = Fk (,) φ = ∫
0 1 k 2 sin 2 θ = ∫ 0
( 1 − 2 )( 1 k − 22 )
where f = am u is called the amplitude of u and x = sin f, and where here and below 0 < k < 1.
Complete Elliptic Integral of the First Kind
35.2. K = Fk (,/) π 2 = ∫ 0 =
1 − k 2 sin 2 θ 0 ∫ ( 1 − 2 )( 1 − k 22 )
Incomplete Elliptic Integral of the Second Kind
35.3. Ek (,) φ = 0 1 − k 2 sin 2 d ∫ − θθ = d ∫ 0
1 k 22
Complete Elliptic Integral of the Second Kind
35.4. E = Ek (,/) π 2 = ∫ 0 1 − k 2 sin 2 θθ d = 0 d ∫
Incomplete Elliptic Integral of the Third Kind
35.5. Π( , , ) kn φ = ∫ 0 =
( 1 n sin ) 2 1 k 2 sin 2 ∫ + 0 θ − θ ( 1 + n 2 )( 1 − 2 )( 1 − k 22 )
ELLIPTIC FUNCTIONS
Complete Elliptic Integral of the Third Kind
35.6. Π( , , / ) kn π 2 = ∫ 0 =
( 1 + n sin ) 2 θ 1 − k 2 sin 2 ∫ 0 ( 1 n + 2 )( θ 2 11 − )( 1 − 22 k )
Landen’s Transformation
sin 2 φ
35.7. tan φ 1 sin( 2 =+ )
k cos 2 or k sin φ =
This yields
35.8. Fk (,)
1 − 2 k sin 2 θ =+ 1 k ∫ 0 1 k 2 − 2 1 sin θ 1
where k 1 = 2 k /( 1 + k ). By successive applications, sequences k k k 1 , , , … and φ φ φ 2 3 1 , 2 , , … are obtained such 3
that k < k 1 < k 2 < k 3 < ⋯ < 1 where lim n →∞ k n = 1 It follows that .
∫ 0 1 sin 2 θ = k
35.10. k 1 1 k
k 2 , … and
Φ n →∞ φ n
The result is used in the approximate evaluation of F(k, f).
Jacobi’s Elliptic Functions From 35.1 we define the following elliptic functions:
35.11. x = sin ( am u ) = sn u
35.12. 1 − x 2 = cos ( am u ) = cn u
35.13. 1 − kx 22 = 1 − k 2 sn 2 u = dn u
We can also define the inverse functions sn − 1 x , cn − 1 x , dn − 1 x and the following:
1 sn u
cn u
35.14. ns u =
35.17. sc u =
35.20. cs u =
35.15. nc u =
35.18. sd u =
35.21. dc u =
35.16. nd u = dn u
35.19. cd u =
35.22. ds u =
dn u
dn u
Addition Formulas
sn cn dn u cn sn dn u u
35.23. sn ( u + )
1 − k 2 sn 2 u s nn 2
ELLIPTIC FUNCTIONS
cn cn u sn sn dn dn u u
35.24. cn ( u )
1 k 2 sn 2 u sn − 2 dn dn u 2 − k sn sn cn cn u u
35.25. dn ( u + ) =
1 k − 2 sn 2 u sn 2
Derivatives
35.26. sn u = cn dn u u
35.28. dn u =− k 2 sn cn u u
du
du
35.27. cn u =− sn dn u u
35.29. sc u = dc nc u u
du
du
Series Expansions
) u ++ ( 1 14
35.30. sn u =−+ u ( 1 k 2 u
k 2 + k 4 ) −+ ( 1 135 k 2 + 1 335 k 4
35.31. cn u =− 1
2 ! ++ ( 14 k 2 )
−+ ( 1 44 k 2 4 16 + k 4 )
35.32. dn u =− 1 k 2 + k 2 ( 4 + k 2 )
− k 2 ( + 16 44 k 2 + k 4 ) u 2 ⋯ ! 4 ! 6 !! +
Catalan’s Constant
35.33. 2 ∫ 0 K dk =
1 1 1 1 π / 2 θ d dk
k = 0 ∫ θ = 0 = 2 − 2 + 2 − = . 915965 1 5594 − k 2 sin 2 θ 1 3 5
Periods of Elliptic Functions Let
35.34. K
K ′=
∫ 0 1 −′ k 2 sin 2 θ
35.35. sn u has periods 4K and 2iK ′
35.36. cn u has periods 4K and 2K + 2iK ′
35.37. dn u has periods 2K and 4iK ′
ELLIPTIC FUNCTIONS
Identities Involving Elliptic Functions
35.38. sn 2 u + cn 2 u = 1 dn 2 u + k 2 sn 2 u = 35.39. 1
35.40. dn u − k cn u = ′ where ′ = k 2 k 1 − k 2 2 35.41. sn − u = 1
1 cn 2 u
+ dn 2 u
35.42. cn u +
35.45. 1 − dn 2 u k sn cn u u
Special Values
35.46. sn 0 = 0 35.47. cn 0 = 1 35.48. dn 0 = 1 35.49. sc 0 = 0 35.50. am 0 = 0
Integrals
35.51. sn u du = ln( dn k ∫ u − k cn u )
35.52. cn ∫ u du = cos ( − 1 k dn u )
35.53. dn u du = sin ( − 1 sn u ∫ )
35.54. sc ∫ u du =
2 ln dc ( u + 1 − k
2 nc u
35.55. cs ∫ u du = ln( ns u − ds u )
35.56. cd u du = ln ( nd u + k sd u ∫ ) k
35.57. dc u du = ln( nc u + s c u ∫ )
sin ( − 1 k cd u ∫ )
35.58. sd u du
35.59. ds u du = ln( ns u − cs u ∫ )
35.60. ns u du = ln ( ds ∫ u − cs u )
sc u
35.61. nc u du
2 ln dc u 1 + − k ⎝⎜ 1 − k 2 ⎠⎟
cos ( − 1 cd u ∫ )
35.62. nd u du =
ELLIPTIC FUNCTIONS
Legendre’s Relation
35.63. EK ′+′− EK KK ′ = π /2 where
35.64. E =
1 k ∫ sin
θθ d K = ∫ 0
1 k − 2 sin 2 θ
k 2 2 π 2 35.65. d sin E θ d K
1 2 −′ 2 k sin θ
MISCELLANEOUS and RIEMANN ZETA FUNCTIONS
Error Function erf ( ) x =
e − du
36.1. erf ( ) x ⎛
36.2. erf ( ) ~ x 1 −
2 x 2 ⎝⎜ + ( 2 x 22 ) − ( 2 x 23 ) ++ ⎠⎟
36.3. erf ( −=− x ) erf ( ), x erf ( ) 0 = erf ( ) 0 , ∞=1
e Complementary Error Function u =− x = − du
erfc ( ) x
1 erf ( )
36.4. erfc ( ) x =− 1 x −
36.5. erfc ( ) ~ x
36.6. erfc ( ) 0 = erfc( ) 1 , ∞=0
Exponential Integral Ei ( ) x = ∫ x du
36.7. Ei ( ) x =−− γ ln x
∫ du 0
36.8. Ei ( ) x ln x ⎛ x
36.9. Ei ( ) ~ x ⎛ 1 − + −
x sin u Sine Integral Si ( ) x = 0 ∫ du u
36.11. Si ( ) x = 11 i ! − 33 i ! +
i ! − 77 i ! +
⎞ cos x ⎛ 2 !! 4 36.12. ! Si ( ) ~ x ⋯ ⋯ ⎞
36.13. Si ( −=− x ) Si ( ), x Si ( ) 0 = Si ( ) 0 , ∞= π2 /
MISCELLANEOUS AND RIEMANN ZETA FUNCTIONS
Cosine Integral Ci( ) x = x u du ∫
∞ cos u
x 1 cos u
36.14. Ci ( ) x =−− ln x
∫ du 0
36.15. Ci ( ) x =−− γ ln x +
i ! + 66 i ! − 88 i ! +
36.16. Ci ( ) ~ x ⎛
Fresnel Sine Integral Sx ()
0 sin u du =π 2 ∫
1 13 i 1357 i i i
⋯ ⎞ (sin ) x 2 ⎛
36.19. Sx ()~
⎨ (cos ) x 2
2 x 3 − 2 3 ⎠⎟ + 7 ⎝⎜ x + ⎠⎟ ⎬ ⎭
⎝⎝⎜ x 2 2 x 5