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

Transcription

1 InverseJacobiCS Notations Traditional nae Inverse of the Jacobi elliptic function cs Traditional notation cs Matheatica StandardFor notation InverseJacobiCS, Priary definition csw ; w cs cs t t t ; Specific values Specialied values For fixed cs 0 cot cs F sinh cs csch For fixed cs K F sinh

2 cs F sinh F sinh cs 0 K cs F sinh cs F sinh K cs K cs K Values at infinities cs cs cs cs K General characteristics Doain and analyticity cs is an analytical function of and which is defined over cs Syetries and periodicities Mirror syetry cs cs Quasi-reflection syetry

3 cs cs F sinh Poles and essential singularities With respect to The function cs does not have poles and essential singularities with respect to ing cs With respect to The function cs does not have poles and essential singularities with respect to ing cs Branch points With respect to For fixed, the function cs has two branch points:, cs, cs, log cs, log With respect to For fixed, the function cs has five branch points: ±, ±, cs,,,, cs, cs, cs, cs,

4 cs, log Branch cuts Branch cut locations: coplicated Series representations Generalied power series Expansions at cs K K ; 0 40 cs K k K k F k 0 k k, k; k k; cs K K O Expansions at cot cot cs cot ; k cs k k k F k, k ; k 3 ; k 0 k cs k k j j k 0 k cot j j k ; cs j jk k jk j k k k 0 j 0 j k 3 jk

5 cs F , ;; ; 3 ;; ; cs cot O, Integral representations On the real axis Of the direct function cs t t t ; cs cs 0 ndcs 0 t t t ; Τ,Τ,0Τ IΤ Τ IΤ Τ cs ndcs t t t ; Τ,Τ,0Τ I tan Π Τ I tan Π Τ 0 tan Π Τ 0 tan Π Τ 0 0 Differential equations Ordinary nonlinear differential equations w 3 w 0 ; w cs Transforations Transforations and arguent siplifications Arguent involving basic arithetic operations

6 cs cs F sinh Identities Functional identities csw w csw w 0 ; w cs Differentiation Low-order differentiation With respect to cs ndcs cs ; cs ndcs cs ndcs With respect to cs Eacs cs ndcs cs E E sinh K ;

7 cs 4 3 cs 4 Eacs Facs ndcs cs Eacs 7 Facs 5 3 cs ndcs Sybolic differentiation With respect to n cs n ndcs n j n j nj n cs j 0 n j jn k 0 j k k jk k kj ; n n cs n cs n ndcs n j jn jn j n nj j F n j, j; j; j 0 ; n n cs n cs n ndcs n n ; n

8 n cs n Π n n ndcs n j jn n j 0 j n j j j n 3 F j, j ; j; j n F, j n ; j n 3 ; ; n With respect to n cs n Π n F n n n n ;, n ; n 3 ;, ; n Fractional integro-differentiation With respect to Α cs Α Α Α K K Α Π F 0 0, ; ; ;, 3Α, Α ;;; ; 0 With respect to Α cs Α Π Α F 0 ; ;, ; 3 ;; Α;, ; 0 Integration Indefinite integration Involving only one direct function cs cs logdscs nscs Representations through ore general functions Through hypergeoetric functions of two variables cs F 0 0, ;; ; 3 ;; ;, Through other functions

9 9 Involving soe hypergeoetric-type functions cs F ;, ; 3 ;, ; Representations through equivalent functions With inverse function cscs With related functions Involving cd cs K cd Involving cn cs cn ; 0 0 Involving dc cs dc K ; 0 0 Involving dn cs dn K ; 0 0 Involving ds cs ds ; 0 Involving nc cs K nc ; Involving nd

10 cs nd K ; 0 Involving ns cs ns Involving sc cs sc Involving sd cs sd ; 0 Involving sn cs sn ; 0 Involving elliptic integrals cs F sinh ; cs K K F sinh ; cs cs 0 ndcs F sinh 0 F sinh 0 0 ; Τ,Τ,0Τ IΤ Τ IΤ Τ 0 0 0

11 ndcs cs F csch ; Τ,Τ,0Τ I tan Π Τ I tan Π Τ Involving other related functions tan Π Τ 0 tan Π Τ 0 0 cs elog, ; a, b ; a, b,,, 3 a b 0 0 History N. H. Abel (86) A. G. Greenhill (89) L. M. Milne Thopson (948)

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