InverseJacobiCS Notations Traditional nae Inverse of the Jacobi elliptic function cs Traditional notation cs Matheatica StandardFor notation InverseJacobiCS, Priary definition 09.39.0.000.0 csw ; w cs 09.39.0.000.0 cs t t t ; Specific values Specialied values For fixed 09.39.03.000.0 cs 0 cot cs 09.39.03.000.0 F sinh 09.39.03.0003.0 cs csch For fixed 09.39.03.0004.0 cs K F sinh
http://functions.wolfra.co cs 09.39.03.0005.0 F sinh F sinh 09.39.03.0006.0 cs 0 K 09.39.03.0007.0 cs F sinh 09.39.03.0008.0 cs F sinh K 09.39.03.0009.0 cs K 09.39.03.000.0 cs K Values at infinities 09.39.03.00.0 cs 0 09.39.03.00.0 cs 0 09.39.03.003.0 cs 0 09.39.03.004.0 cs K General characteristics Doain and analyticity cs is an analytical function of and which is defined over. 09.39.04.000.0 cs Syetries and periodicities Mirror syetry 09.39.04.000.0 cs cs Quasi-reflection syetry
http://functions.wolfra.co 3 09.39.04.0003.0 cs cs F sinh Poles and essential singularities With respect to The function cs does not have poles and essential singularities with respect to. 09.39.04.0004.0 ing cs With respect to The function cs does not have poles and essential singularities with respect to. 09.39.04.0005.0 ing cs Branch points With respect to For fixed, the function cs has two branch points:,. 09.39.04.0006.0 cs, 09.39.04.0007.0 cs, log 09.39.04.0008.0 cs, log With respect to For fixed, the function cs has five branch points: ±, ±,. 09.39.04.0009.0 cs,,,, 09.39.04.000.0 cs, 09.39.04.00.0 cs, 09.39.04.00.0 cs, 09.39.04.003.0 cs,
http://functions.wolfra.co 4 09.39.04.004.0 cs, log Branch cuts Branch cut locations: coplicated Series representations Generalied power series Expansions at 0 09.39.06.000.0 cs 09.39.06.000.0 K K 8 8 3 6 4 ; 0 40 cs K k K k F k 0 k k, k; k k; 09.39.06.0007.0 cs K K O Expansions at 0 09.39.06.0003.0 cot 3 3 3 5 3 cot cs cot ; 0 4 64 09.39.06.0004.0 k cs k k k F k, k ; k 3 ; k 0 k 09.39.06.0008.0 cs k k j j k 0 k cot j j k ; 09.39.06.0005.0 cs j jk k jk j k k k 0 j 0 j k 3 jk
http://functions.wolfra.co 5 09.39.06.0006.0 cs F 0 0 09.39.06.0009.0, ;; ; 3 ;; ; cs cot O, Integral representations On the real axis Of the direct function 09.39.07.000.0 cs t t t ; 09.39.07.000.0 cs cs 0 ndcs 0 t t t ; Τ,Τ,0Τ IΤ 0 0 0 Τ 0 0 0 IΤ 0 0 0 Τ 0 0 0 09.39.07.0003.0 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 09.39.3.000.0 w 3 w 0 ; w cs Transforations Transforations and arguent siplifications Arguent involving basic arithetic operations
http://functions.wolfra.co 6 09.39.6.000.0 cs cs F sinh Identities Functional identities 09.39.7.000.0 csw w 4 4 4 csw w 0 ; w cs Differentiation Low-order differentiation With respect to cs 09.39.0.000.0 ndcs 09.39.0.000.0 cs ; 09.39.0.0003.0 cs ndcs 09.39.0.00.0 cs ndcs With respect to cs 09.39.0.0004.0 Eacs cs ndcs 09.39.0.0005.0 cs E E sinh K ;
http://functions.wolfra.co 7 09.39.0.0006.0 cs 4 3 cs 4 Eacs Facs 3 4 5 ndcs 09.39.0.00.0 3 cs 3 4 3 3 3 8 Eacs 7 Facs 5 3 cs 3 5 4 5 4 5 7 5 3 3 5 ndcs Sybolic differentiation With respect to 09.39.0.003.0 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 09.39.0.004.0 n cs n cs n ndcs n j jn jn j n nj j F n j, j; j; j 0 ; n 09.39.0.005.0 n cs n cs n ndcs n n ; n
http://functions.wolfra.co 8 09.39.0.0007.0 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 09.39.0.0008.0 n cs n Π n F n n n n ;, n ; n 3 ;, ; n Fractional integro-differentiation With respect to 09.39.0.0009.0 Α cs Α Α Α K K Α Π F 0 0, ; ; ;, 3Α, Α ;;; ; 0 With respect to 09.39.0.000.0 Α cs Α Π Α F 0 ; ;, ; 3 ;; Α;, ; 0 Integration Indefinite integration Involving only one direct function 09.39..000.0 cs cs logdscs nscs Representations through ore general functions Through hypergeoetric functions of two variables 09.39.6.000.0 cs F 0 0, ;; ; 3 ;; ;, Through other functions
http://functions.wolfra.co 9 Involving soe hypergeoetric-type functions 09.39.6.000.0 cs F ;, ; 3 ;, ; Representations through equivalent functions With inverse function 09.39.7.000.0 cscs With related functions Involving cd 09.39.7.000.0 cs K cd Involving cn 09.39.7.0003.0 cs cn ; 0 0 Involving dc 09.39.7.0004.0 cs dc K ; 0 0 Involving dn 09.39.7.0005.0 cs dn K ; 0 0 Involving ds 09.39.7.0006.0 cs ds ; 0 Involving nc 09.39.7.0007.0 cs K nc ; Involving nd
http://functions.wolfra.co 0 09.39.7.0008.0 cs nd K ; 0 Involving ns 09.39.7.0009.0 cs ns Involving sc 09.39.7.000.0 cs sc Involving sd 09.39.7.00.0 cs sd ; 0 Involving sn 09.39.7.00.0 cs sn ; 0 Involving elliptic integrals 09.39.7.003.0 cs F sinh ; 0 09.39.7.004.0 cs K K F sinh ; 0 09.39.7.006.0 cs cs 0 ndcs F sinh 0 F sinh 0 0 ; Τ,Τ,0Τ IΤ 0 0 0 Τ 0 0 0 IΤ 0 0 0 Τ 0 0 0
http://functions.wolfra.co 09.39.7.007.0 ndcs cs F csch ; Τ,Τ,0Τ I tan Π Τ I tan Π Τ Involving other related functions 09.39.7.005.0 0 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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