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