EllipticK. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

Transcription

1 EllipticK Notations Traditional name Complete elliptic integral of the first kind Traditional notation K Mathematica StandardForm notation EllipticK Primary definition K F Specific values Values at fixed points K K K K K

2 Values at infinities K K K K K 0 Singular values K K ; K ; K K ; K K ; K K 5 ; 5 K K 6 ; 6 K K 7 ; K K 8 ; K

3 K 9 ; K K 0 ; K K ; K K ; K K ; K K K ; ; K 5 ; K K 6 ; K K 7 ; K

4 K 8 ; K K K 9 ; K 0 ; K ; General characteristics Domain and analyticity K is an analytical function of which is defined over the whole complex -plane K Symmetries and periodicities Mirror symmetry K K ;, Periodicity The function K is not periodic. Poles and essential singularities The function K does not have poles and essential singularities ing K Branch points The function K has two branch points:, K, K, log

5 K, log Branch cuts The function K is a single-valued function on the -plane cut along the interval,. The function K is continuous from below on the interval, K,, lim Kx Ε Kx ; x Ε lim Kx Ε K x Kx ; x Ε0 Series representations Generalied power series Expansions at generic point 0 For the function itself K K 0 K 0 G,,, 0 0, arg arg 0 arg 0 arg 0 K 0 0 E 0 0 G,,, 0 0, 0 0 arg 0 arg 0 K 0 0 E 0 K 0 0 E 0 0 ; K K 0 K 0 G,,, 0 0, arg arg 0 arg 0 arg 0 K 0 0 E 0 0 G,,, 0 0, 0 0 arg 0 arg 0 K 0 0 E 0 K 0 0 E 0 0 O 0

6 K k G,, k, k 0 0, k k 0 k arg 0 arg 0 k F k, k ; k ; 0 0 k K k F k k 0 k, k ; k ; 0 0 k K K 0 K 0 arg 0 arg 0 O 0 Expansions on branch cuts For the function itself K Kx K x argx x x K x x E x argx G,, x, 0, x x x argx ; x x x K x x E x K x x E x G,, x, 0, x K Kx K x argx x x K x x E x argx G,, x, 0, x x x argx K x x E x K x x E x G,, x, 0, x O x ; x x K k 0 k G,, x k, k 0, k k argx k F k, k ; k ; x xk ; x x K Kx K x Expansions at 0 argx O x ; x x For the function itself

7 7 K K K k ; O k k k k ; K F, ; ; ; K O K F ; F n k 0 k n k k k n n K F, n n, n ; n, n ; n Summed form of the truncated series expansion. Expansions at For the function itself K log 9 6 log log 8 6 log 7 ; K log 9 6 O log log 8 6 log 7 O K k log k k k k log k k 0 k k 0 k 9 k 5 6 k k 0 k k k k k i k i k ;

8 K k log k k k k Ψk Ψ k k 0 k k 0 k k ; log k K K k 0 k Ψk Ψ k k ; K log O log O K F ; F n k 0 k n k log Ψk Ψ k k K G,, n, n,, n, n, n Summed form of the truncated series expansion. Expansions at For the function itself K log 9 log 6 log 7 log ; K log 9 O log 6 log 7 log O k K log k k 0 k k k 0 k k k k k i k i k k k log k 0 k i ; i K log k k 0 k k k k 0 k Ψk Ψ k k ;

9 K log K k k 0 k Ψk Ψ k k ; K log O log O K log log arg 0 ; True K F ; F n m k k 0 k log Ψk Ψ k Ψ k k K G,, m, m,, 0, m, m, 0 m Summed form of the truncated series expansion. Residue representations K j 0 res s s s s j ; s K s s res s s s j 0 j ; Other series representations K q k k K k q k q k Integral representations On the real axis

10 0 Of the direct function K sin t t ; arg K 0 t t t ; arg K t t t ; arg Contour integral representations K s s s s s Differential equations Ordinary linear differential equations and wronskians For the direct function itself w w w 0 ; w c K c K W K, K w g g g g g g g w g g w 0 ; w c Kg c K g g W Kg, g g g w g g h g g g h g w g g g g h g g g h 0 ; w c h Kg c h K g h h h g h g h h w

11 W h Kg, h K g h g g g a r w a r r s a r w a r r a r s r s a r w 0 ; w c s Ka r c s K a r s r W s Ka r, s K a r a r w a r logr logs w a r logr logs log s w 0 ; a r w c s Ka r c s K a r W s Ka r, s K a r s logr a r Identities Functional identities K K K K ; arg ; K K K KK K K KK K K K K K

12 K K Complex characteristics Real part ReKx y F 0 0,, 5, 5 ;;; y, x x, ; ; ; 8 F 0 0,,, ;;;, ; ; ; y, x ; x y ReK x cos x K sin x Ksin x sin x ; x 0 x Imaginary part ImKx y y F, 5,, 5 ;;; y, x 9 x y, ; ; ; 6 F 0 0 5, 7, 7, 5 ;;;, ; ; ; y, x ; x y ImK x cos x Ksin x K sin x sin x ; x 0 x Differentiation Low-order differentiation K E K K E 5 K Symbolic differentiation n K n n n F n, n ; n ; ; n n K n F n, ; n; ; n

13 Fractional integro-differentiation Α K Α F Α, ; Α; Integration Indefinite integration Involving only one direct function Ea a Ka Ka a K E K Involving one direct function and elementary functions Involving power function Involving power Linear argument Α Ka Α Α F,, Α;, Α ; a Α K Α Α F,, Α;, Α ; a Ea a a Ka Ka 9 a Ka F,, ;, ; a Ka 8 a F,,, ;,, ; a 8 log loga Ka 8 9 a F,, 5, 5 ;,, ; a 6 a log a loga

14 Power arguments Α Ka r Α Α F,, Α r ;, Α ; a r r Ea a Ka Ka a a Ea a a Ka Ka 9 a Ka Ka F 5 a 9 a 6 a 6 Ea 5 a 6 a 6 a 6 Ka,, ;, ; a Ka 6 a F,,, ;,, ; a 8 log loga Ka Ka Ea a F,, 5, 5 ;,, ; a 6 a log a loga Involving algebraic functions E E K Definite integration For the direct function itself t Α Ktt 0 Α F,, Α;, Α ; ; ReΑ t Α Α Α Ktt ; 0 ReΑ 0 Α

15 a x a a x b b x K b x b b b x x a a b b sech Α Ksech Α Ktanh Α ; Rea Reb 0 cosh b a a a b b Representations through more general functions Through hypergeometric functions Involving F K F, ; ; Through Meijer G Classical cases for the direct function itself K G,,, Classical cases involving algebraic functions K , G,, ;, K, G,, ;, K, G,, ;, K G,,,, ;, 0

16 K G,,, K G,,, K G,,, K, G,, K, G, 5, 5,, K, G,,,, K, G,, ;, K, G,, ;, K, G,, ;, K, G,, ;, 0 K G,,,,, 0

17 K, G,,,, K G,,, K G,,, 0, ;, K G,,, 0, ;, K G,,, K, G,, K, G,, K G,,, K, G,, ;, K, G,, ;, K G,,, 0, ;, 0

18 K, G,, ; Re K, G,, ; Re K G,,, ; Re K, G,, ; Re K G,,,, ; Re K G,,,, ; Re K G,,, ; Re K G,, ; Re K G,,, K G,,,, ;, 0

19 K G,, K G,, ,, ;, 0 ;, 0 K G,,, ;, K G,,, ;, K G,,,, K G,,, K G,,,, K G,,, Classical cases involving unit step Θ Θ K G,0, Θ K G 0,, Θ K G,0,,,,, ;, 0

20 Θ K G 0,,,, ;, Θ K G,0,, 0, ;, 0 Θ K G 0,,, 0, Θ K Θ K G,0, G 0,,,,,, ;, 0 Θ K G,0,, ;, Θ K G 0,,, Θ K G,0,, 0, ;, Θ K G 0,,, 0, Θ K G,0,,, ;, Θ K G 0,,,,

21 Θ K G,0,, ;, Θ K G 0,,, Θ K G,0,, 5, ;, Θ K G,0,, ;, Θ K G 0,,, Θ K G 0,,, 5, ;, Θ K G,0,, ;, Θ K G 0,,, ;, Θ K G,,0, ; Re Θ K G,,0, ; Re Θ K G,0,, 5, ;, 0

22 Θ K G 0,,, 0, ;, Θ K G 0,,, 0, ;, Θ K G,0,, 0, ;, Θ K G 0,,, 0, ;, Θ K G,0,, 0, ;, Θ K G,0,, 0, ;, Θ K G 0,,, 0, Θ K G 0,,, ; Re Θ K G 0,,, ; Re 0 Classical cases involving sgn sgn K G,,, Classical cases involving powers of complete elliptic integral K

23 K , G,,,, K, G,,,,, ;, K 8 G,,,,, 0 ;, K, 8 G,,,, 0 ;, 0 Generalied cases involving algebraic functions K G,,,,, ;, 0 K G,,, K G,,,,, K G,,,,, ; Re K G,,,,, ; Re K G,,,, 0, ; Re K G,,,, 0, ; Re 0

24 K G,,,, ; Re K G,,,, ; Re K G,,,, ; Re 0 0, K G,,,, K G,,,, K G,,,, K G,,,, K G,,,, ;,, K G,,,, ;,,

25 K G,,,,, ; Re K G,,,, ; Re K G,,,, ; Re K G,,,, ; Re K G,,,, ; Re 0 Generalied cases involving unit step Θ Θ K G,0,,, 0, Θ K Θ K G,0,, G 0,,,,,, 0, Θ K G 0,,,,, Θ K G,0,,, 0,

26 Θ K G,0,,, ;, Θ K G 0,,,, 0, ;, Θ K G 0,,,, Θ K Θ K G,0,, G 0,,, Θ K G,0,, Θ K G 0,,, ,,,,,, ;, 0 ;, Θ K G 0,,,, 0, ; Re Θ K G 0,,,, ; Re 0 0, Θ K G,0,,, ; Re 0 0, Θ K G 0,,,, ; Re 0 0,

27 Θ K G,0,,, 0, ; Re Θ K G,0,,, 0, ; Re Θ K G 0,,, 0, ; Re 0 Generalied cases involving sgn sgn K G,,,, Generalied cases involving powers of complete elliptic integral K K G,,,,,,, ; Re 0 K G,,,,,, 0 ; Re 0 Through other functions Involving incomplete elliptic integrals K 0; K K F K F sin

28 8 Involving elliptic theta functions Km ϑ 0, qm Involving inverse Jacobi functions K sn K dn K cn 0 ; Involving some elliptic-type functions K agm, K q exp Ω Ω e e Ω ; e, e, e Ω ; g, g, Ω Ω ; g, g, Ω ; g, g g, g g Ω, Ω, g Ω, Ω K Ω ; q exp Ω K Ω Ω Ω, Ω Ω g, g, Ω g, g Involving Legendre functions K P K Q Involving some hypergeometric-type functions K F ;, ; ;, Representations through equivalent functions With inverse function amkm m

29 9 With related functions E K K K E K K E Theorems The period T of a mathematical pendulum in a gravitational field The period T of a mathematical pendulum of length l in a gravitational field with acceleration g and maximal angle of excursion Α is given by T l g Ksin Α. The partition function for a one-dimensional monatomic ideal classical gas The partition function Z for a one-dimensional monatomic ideal classical gas of n atoms in a box of length l at temperature T is given by Z n n Kq exp ΛT wavelength. The magnetic induction of an infiniely long selenoid 8 l n, where ΛT is the thermal de Broglie The magnetic induction B of an infinitely long solenoid formed by a wire (parametried by Φ) R cosφ, R sinφ, R Φ tanα carrying the current i 0 is at the center line is given by B 0, i 0 cotα K 0 cotα K cotα, i 0 cotα R. The lattice Green function for the body-centered cubic lattice The lattice Green function Gε ε cosxcosycos x y for the simple cubic lattice can be expressed as Α Α K Β Β Β Β K Β Β Β Β ; Α ε 6 ε ε 9 Β Α Α.

30 0 The probability that a random walk in three dimensions will return to its origin The probability p 0 that a random walk in three dimensions will return to its point of origin is given by p K History A. M. Legendre (8, 85) C. G. J. Jacobi (89)

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