JacobiDS Notations Traditional name Traditional notation Mathematica StandardForm notation Primary definition
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1 JacobiDS Notations Traditional name Jacobi elliptic function ds Traditional notation dsz m Mathematica StandardForm notation JacobiDSz, m Primary definition dnz m snz m Specific values Specialized values For fixed z Case m dsz 0 cscz ds z Π 2 0 secz ds z Π k 0 csc z Π k 2 2 ; k Case m 1
2 ds z Π 1 sechz ds z Π k 1 csch z Π k 2 2 ; k For fixed m Values at quarter-period points in the fundamental period parallelogram ds0 m dskm m m ds2 Km m ds3 Km m ds4 Km m 1 m ds K1 m m m ds2 K1 m m ds3 K1 m m m ds4 K1 m m dskm K1 m m ds2 Km K1 m m m ds3 Km K1 m m ds4 Km K1 m m dskm 2 K1 m m m 1 m ds2 Km 2 K1 m m ds3 Km 2 K1 m m 1 m
3 ds4 Km 2 K1 m m ds2 r Km 2s K1 m m ; r, s Values at half-quarter-period points ds Km m 4 1 m 1 1 m ds K1 m m 2 4 m 1 m ds Km K1 m 2 m m 1 1 m 1 m 4 m 1 m 1 m General characteristics Domain and analyticity dsz m is a meromorphic function of z and m which is defined over z mdsz m Symmetries and periodicities Parity dsz m is an odd function with respect to z dsz m dsz m Mirror symmetry dsz m dsz m Periodicity dsz m is a doubly periodic function with respect to z with periods 4 K1 m and 4 Km dsz 2 Km m dsz m dsz 4 Km m dsz m
4 dsz 2 K1 m m dsz m dsz 4 K1 m m dsz m dsz 2 Km 2 K1 m m dsz m dsz 2 s K1 m 2 r Km m 1 rs ; r, s Poles and essential singularities With respect to z For fixed m, the function dsz m has an infinite set of singular points: a) z 2r Km 2s K1 m, r, s, are the simple poles with residues 1 rs ; b) z is an essential singular point ing z dsz m 2s K1 m 2r Km, 1 ; r, s,, res z dsz m2s K1 m 2r Km 1 rs ; r, s Branch points With respect to m For fixed z, the function dsz m is a meromorphic function in m that has no branch points m dsz m P. Walker With respect to z For fixed m, the function dsz m does not have branch points z dsz m Branch cuts With respect to m For fixed z, the function dsz m is a meromorphic function in m that has no branch cuts m dsz m P. Walker With respect to z
5 5 For fixed m, the function dsz m does not have branch cuts z dsz m Series representations Generalized power series Expansions at z dsz m 1 z m z m 8 m2 z 3 ; z dsz m 1 z m z m 8 m2 z m 48 m 2 32 m 3 z m 96 m m m 4 z m 1072 m m m m 5 z 9 Oz k j 1 1 kj dn kj m j 2 k 2 j k 0 j 0 r 0 1 r r 1 j r q r,j z 2 k1 ; q j,0 1 q j,k 1 k j i i k 1 i sn i m q j,ki k i 1 2 i 1 n n k sn 0 m 1 sn n m j 0 k 0 2 n 2 j cn j m dn k m jkn cn 0 m 1 n1 n1 cn n m j 0 k 0 n1 n1 2 n 1 2 j 1 sn jm dn k m jkn1 dn 0 m 1 dn n m m j 0 k 0 2 n 1 2 j 1 sn jm cn k m jkn dsz m 1 z 1 Oz2 Expansions at z 2 r Km 2 s K1 m dsz m 1 rs 1 z z m z z 0 1 z z 0 z 0 2 r Km 2 s K1 m r s m2 8 m 7 z z 0 3 ;
6 k j 1 1 kj dn kj m j 1 rs 2 k 2 j k 0 j 0 r 0 1 r r 1 j r q r,j z z 0 2 k1 ; z 0 2 r Km 2 s K1 m r s q j,0 1 q j,k 1 k j i i k 1 i sn i m q j,ki k i 1 2 i 1 n n k sn 0 m 1 sn n m j 0 k 0 2 n 2 j cn j m dn k m jkn cn 0 m 1 n1 n1 cn n m j 0 k 0 n1 n1 2 n 1 2 j 1 sn jm dn k m jkn1 dn 0 m 1 dn n m m j 0 k 0 2 n 1 2 j 1 sn jm cn k m jkn dsz m 1rs 1 Oz z 0 2 ; z 0 2 r Km 2 s K1 m r s z z 0 Expansions at m dsz m cscz z cosz 7 sinz sin3 z csc2 zm z2 12 sin2 z z 4 sin4 z z 8 z 2 4 cos2 z 3 cos4 z 29 csc 3 zm 2 ; m
7
8 z z z z z z sin2 z z z z z z sin4 z z z z z z sin6 z z z z z z sin8 z z z z z z sin10 z z z z z z sin12 z z z z z sin14 z z z z sin16 z z z sin18 z z sin20 z z z z z z z cos2 z z z z z z cos4 z z z z z z cos6 z z z z z z cos8 z 4096 z z z z z cos10 z z z z z cos12 z z z z cos14 z z z cos16 z z cos18 z cos20 z csc 11 zm 10 Om dsz m cscz1 Om Expansions at m dsz m cschz z coshz 7 sinhz sinh3 z csch2 zm z2 12 sinh2 z z 4 sinh4 z z 8 z 2 4 cosh2 z 3 cosh4 z 29 csch 3 zm 1 2 ; m
9 9
10 z 9728 z z z cosh11 z z z z cosh13 z z 3400 z cosh15 z z cosh17 z z z z z sinhz z z z z sinh3 z z z z z sinh5 z z z z z sinh7 z z z z z sinh9 z z z z z sinh11 z z z z sinh13 z z z sinh15 z z sinh17 z 2835 sinh19 z csch 10 1 zm z z z z z z z z z sinh2 z z z z z z sinh4 z z z z z z sinh6 z z z z z z sinh8 z z z z z z sinh10 z z z z z z sinh12 z z z z z sinh14 z z z z sinh16 z z z sinh18 z z sinh20 z z z z z z z cosh2 z z z z z z cosh4 z z z z z z cosh6 z z z z z z cosh8 z 4096 z z z z z cosh10 z z z z z cosh12 z z z z cosh14 z z z cosh16 z z cosh18 z cosh20 z csch 11 zm 1 10 Om dsz m cschz1 Om 1 q-series Π 2 Km csc Π z 2 Km 2 Π Km sin k 0 qm 2 k1 1 qm 2 k1 2 k 1 Π z 2 Km Other series representations
11 Π 2 K1 m Km 1 k csch Π K1 m k z k 2 Km rs dsz m O1 ; z 2s K1 m 2r Km r, s z 2s K1 m 2r Km Product representations m 1 m 4 2 qm csc Π z 2 Km k qm 2 k1 cos Π z k2 qm4 Km 1 2 qm 2 k cos Π z k qm4 Km Differential equations Ordinary nonlinear differential equations w z wz 2 wz 2 2 m 1 0 ; wz dsz m Transformations Transformations and argument simplifications Argument involving basic arithmetic operations ds z m dsz 1 m dsz 1 m ds z m ds z 1 m dsz m dsx y m dnx m cny 1 m dny 1 m m snx m cnx m sny 1 m dny 1 m snx m cnx m cny 1 m dnx m sny 1 m ; x, y ds ds ds m nsz m 1 m z m 1 1 m csz m m z m m nsz m 1 m z 1 m 1 m
12 12 ds m 1 csz m m z m m Landen's transformation: ds 1 1 m z 1 1 m 1 1 m m snz m m snz m cnz m Gauss' transformation: ds 1 m z 4 m 1 m snz m 2 1 m 2 1 m snz m n th degree transformations: ds z M l M dsz m r 1 n1 2 r1 Km 2 1 m sn m 2 snz m 2 n 1 ns 2 r Km n m 2 snz m 2 ; n1 n 1 2 l m n sn 2 r 1 2 r 1 Km n m 8 n1 2 sn M r 1 2 r1 Km n sn 2 r Km n m 2 m 2 ds z M Km n M n 1 l l dnz m r m sn 2 r Km n m 2 snz m 2 ; 2 r1 Km 1 ns m 2 snz m 2 n n 2 2 l m n sn r 1 n 2 r 1 Km n m 8 2 sn M n r 1 2 r1 Km n sn 2 r Km n m 2 m 2 Argument involving half-periods dsz Km m 1 m ncz m dsz Km m 1 m ncz m dsz 3 Km m 1 m ncz m dsz 2 r 1 Km m 1 r 1 m ncz m ; r
13 dsz K1 m m dsz K1 m m dsz 3 K1 m m m cnz m m cnz m m cnz m ; s dsz 2 s 1 K1 m m 1 s1 m cnz m ; s dsz Km K1 m m m 1 m sdz m dsz K1 m Km m m 1 m sdz m dsz K1 m Km m m 1 m sdz m dsz K1 m Km m m 1 m sdz m dsz K1 m 3 Km m m 1 m sdz m dsz 4 s 1 K1 m 4 r 1 Km m m 1 m sdz m dsz 4 s 1 K1 m 4 r 1 Km m m 1 m sdz m dsz 4 s 1 K1 m 4 r 1 Km m m 1 m sdz m dsz 4 s 1 K1 m 4 r 1 Km m m 1 m sdz m dsz 2 s 1 K1 m 2 r 1 Km m 1 rs m 1 m sdz m Argument involving inverse Jacobi functions dscd 1 z m m 2 1 m z 2 dscn 1 z m m 2 m z2 m z 2 dscs 1 z m m 2 z 2 m dsdc 1 z m m 2 1 m z2 z 2 1
14 dsdn 1 z m m 2 m z2 1 z dsnc 1 z m m 2 m 1 z2 m 1 z dsnd 1 z m m 2 m z dsns 1 z m m 2 z 2 m dssc 1 z m m 2 1 m 1 z z 2 dssd 1 z m m 1 z dssn 1 z m m 2 1 m z2 Addition formulas z dnu m dnv m m snu m cnu m snv m cnv m dsu v m cnv m dnv m snu m cnu m dnu m snv m dsu v m dsu v m dnv m2 m cnv m 2 snu m 2 snu m 2 snv m 2 Half-angle formulas ds z 2 2 m 1 m dnz m m cnz m 1 cnz m Multiple arguments Double angle formulas ds2 z m dnz m2 m snz m 2 cnz m 2 2 snz m cnz m dnz m Identities
15 15 Functional identities wz 2 wz 2 m 1 wz 2 m w2 z 2 wz 4 m 2 m 2 0 ; wz dsz m Complex characteristics Real part Redsx y m cny 1 m dnx m dny 1 m 2 snx m m cnx m 2 cny 1 m dnx m snx m sny 1 m 2 dny 1 m 2 snx m 2 cnx m 2 cny 1 m 2 dnx m 2 sny 1 m 2 ; x, y, m Imaginary part cnx m dny 1 m cny 1 m 2 dnx m 2 m snx m 2 sny 1 m Imdsx y m ; x, y, m dny 1 m 2 snx m 2 cnx m 2 cny 1 m 2 dnx m 2 sny 1 m 2 Absolute value dsx y m cny 1 m2 dnx m 2 dny 1 m 2 m 2 cnx m 2 snx m 2 sny 1 m 2 dny 1 m 2 snx m 2 cnx m 2 cny 1 m 2 dnx m 2 sny 1 m 2 ; x, y, m Argument argdsx y m tan 1 cny 1 m dnx m dny 1 m 2 snx m m cnx m 2 cny 1 m dnx m snx m sny 1 m 2, cnx m dny 1 m cny 1 m 2 dnx m 2 m snx m 2 sny 1 m ; x, y, m Conjugate value cny 1 m dnx m dny 1 m m cnx m snx m sny 1 m dsx y m ; x, y, m snx m dny 1 m cnx m dnx m cny 1 m sny 1 m Differentiation Low-order differentiation With respect to z dsz m csz m nsz m z
16 dsz m csz m 2 nsz m 2 z 2 With respect to m dsz m 1 csz m nsz m m 1 z Eamz m m m dnz m scz m m 2 m 1 m dsz m 1 m 2 4 m 1 2 m 2 m 1 z Eamz m m m dnz m scz m m 1 z Eamz m m m cdz m snz m csz m 2 2 m 1 nsz m m 1 z Eamz m m m dnz m scz m csz m 2 m nsz m m 1 z Eamz m m m dnz m scz m csz m Eamz m m Famz m m 1 m m nsz m 2 z 2 dnz m scz m m 1 m cnz m scz m m 1 z Eamz m m dnz m scz m snz m m 1 1 dcz m dnz m ncz m m 1 z Eamz m m m cdz m snz m m 1 1 m 1 m m 1 z Eamz m m dnz m m cnz m snz m 1 m snz m2 csz m dsz m nsz m 2 m 1 z Eamz m m m dnz m scz m m 1 z Eamz m m m cdz m snz m Symbolic differentiation With respect to z n dsz m n1 dsz m n z n j 0 n 1 j j csz m z j jn1 nsz m ; n z jn n dsz m 1 k1 2 2 k1 1 B 2 k z n 1 n n z n1 z n1 k 2 k n 1 k 1 Π z 2 Km 2 k 2 1n Π n1 2 k 1 n qm 2 k1 sin Π n Km n1 k 0 qm 2 k k 1 Π z 2 Km ; n Fractional integro-differentiation With respect to z
17 Α dsz m 1 k1 2 2 k1 1 B 2 k z Α exp z, 1z Α1 2 2 k Α 2 k 1 k 1 Π 2 Km 2 k z 2 kα1 2 Α1 Π 52 z 1Α 2 k 1 qm 2 k1 1F Km 2 qm 2 k1 2 1; 1 Α 1 2, 3 Α 2 k 1 2 Π 2 z 2 ; 2 16 Km 2 k 0 Integration Indefinite integration Involving only one direct function cnz m dsz m z log snz m Representations through equivalent functions With inverse function dsds 1 z m m z With related functions Involving am dsz m 2 csc 2 amz m m Involving one other Jacobi elliptic function Involving cd dsz m 2 1 m 1 cdz m 2 Involving cn dsz m 2 m cnz m2 m 1 1 cnz m 2 Involving cs dsz m 2 csz m 2 m 1
18 18 Involving dc dsz m 2 1 m dcz m2 dcz m 2 1 Involving dn dsz m 2 m dnz m2 1 dnz m 2 Involving nc dsz m 2 m 1 ncz m2 m 1 ncz m 2 Involving nd dsz m 2 m ndz m 2 1 Involving ns dsz m 2 nsz m 2 m Involving sc dsz m 2 1 m 1 scz m2 scz m 2 Involving sd sdz m Involving sn dsz m 2 1 m snz m2 snz m 2
19 19 Involving two other Jacobi elliptic functions Involving cd and cs csz m cdz m Involving cd and sc cdz m scz m Involving cs and dc dcz m csz m Involving dc and sc dcz m scz m Involving dn and ns dnz m nsz m Involving dn and sn dnz m snz m m dnz m snz m dnz m 1 dnz m 1 Involving nd and ns nsz m ndz m Involving nd and sn
20 ndz m snz m Involving three other Jacobi elliptic functions cnz m 2 dcz m cnz m 1 cnz m 1 csz m cnz m csz m 2 1 dnz m csz m csz m 2 1 dnz m csz m ncz m dcz m csz m ncz m 1 ncz m dnz m ncz m csz m ncz m 1 ncz m cdz m csz m 2 1 csz m ndz m cnz m csz m 2 1 csz m ndz m dnz m cnz m 1 cnz m 1 nsz m csz m 2 1 dnz m nsz m dcz m ncz m ncz m 1 ncz m 1 nsz m dnz m ncz m 2 ncz m 1 ncz m 1 nsz m dcz m nsz m 1 nsz m 1 cnz m nsz m
21 ncz m nsz m 1 nsz m 1 cdz m nsz m m cnz m dnz m dnz m 2 m 1 scz m cnz m 2 scz m cdz m cnz m 1 cnz m nsz m 1 nsz m 1 scz m cdz m dnz m scz m 2 1 nsz m scz m cnz m dnz m scz m 2 1 scz m dnz m scz m 2 1 ncz m scz m cdz m scz m 2 1 ndz m 2 scz m cnz m scz m 2 1 ndz m scz m csz m 2 1 sdz m ndz m ncz m 2 sdz m ncz m 1 ncz m 1 ndz m dcz m 2 sdz m dcz m ndz m 1 dcz m ndz m scz m 2 1 sdz m ndz m 2 scz m 2
22 csz m 2 1 dnz m snz m dcz m 2 dnz m snz m dcz m dnz m dcz m dnz m dnz m snz m cdz m dnz m 1 cdz m dnz m ncz m snz m cdz m ncz m 1 ncz m dcz m ncz m snz m ncz m 1 ncz m dnz m ncz m2 snz m ncz m 1 ncz m csz m 2 1 snz m ndz m ncz m 2 snz m ncz m 1 ncz m 1 ndz m ndz m snz m cdz m ndz m cdz m ndz m dnz m scz m 2 1 snz m scz m scz m 2 1 snz m ndz m scz m dcz m snz m 1 snz m 1 csz m snz m scz m snz m 1 snz m 1 cdz m snz m 2 Involving four other Jacobi elliptic functions
23 cnz m dcz m cnz m csz m nsz m dcz m cnz m ncz m nsz m dnz m ncz m cnz m ncz m nsz m dcz m dnz m dcz m nsz m csz m dnz m ncz m nsz m scz m cdz m m cnz m m ncz m ncz m dnz m scz m cnz m scz m cdz m cnz m ncz m dnz m scz m cnz m ncz m dnz m scz m cdz m dnz m ncz m cnz m dnz m csz m scz m dnz m csz m scz m ncz m cdz m csz m scz m ndz m cnz m csz m scz m ndz m dnz m csz m scz m nsz m scz m
24 dnz m scz m nsz m scz m cnz m csz m scz m sdz m ndz m 2 scz m csz m dcz m sdz m dcz m ndz m csz m ndz m ncz m sdz m ncz m ndz m dnz m2 cdz m scz m sdz m scz m cdz m scz m sdz m ndz m 2 scz m cnz m ndz m scz m sdz m ndz m 2 scz m cnz m m dnz m scz m sdz m dnz m scz m sdz m dnz m scz m dcz m nsz m snz m cnz m nsz m snz m cdz m 2 dnz m dcz m2 nsz m snz m dnz m ncz m nsz m snz m cdz m dcz m nsz m snz m csz m snz m
25 m 1 dcz m snz m dcz m dnz m cnz m snz m cdz m cdz m dnz m ncz m m dcz m snz m dcz m dnz m ncz m snz m cdz m 2 dnz m ndz m dcz m 2 snz m dcz m 2 ndz m dnz m ncz m snz m ncz m ndz m cdz m dnz m csz m scz m snz m scz m csz m scz m snz m ndz m scz m dnz m cdz m csz m dnz m snz m dcz m snz m dcz m ndz m csz m snz m dnz m csz m dnz m dcz m snz m dcz m csz m ncz m snz m ncz m ndz m cdz m csz m ndz m snz m ndz m cnz m scz m snz m ndz m scz m
26 scz m snz m ncz m cdz m snz m cnz m m scz m snz m scz m snz m dnz m scz m cnz m m dcz m sdz m snz m dcz m sdz m snz m dcz m snz m Involving five other Jacobi elliptic functions cnz m dnz m scz m sdz m ndz m scz m cdz m csz m dnz m snz m ndz m csz m dnz m dcz m snz m dcz m ndz m Involving Weierstrass functions e 1 e 3 Σ 2 Σ z e 1 e 3 ; g 2, g 3 z e 1 e 3 ; g 2, g 3 ; Ω 1, Ω 2, Ω 3 Ω 1 g 2, g 3, Ω 1 g 2, g 3 Ω 3 g 2, g 3, Ω 3 g 2, g 3 m Λ Ω 3 Ω 1 e n Ω n ; g 2, g 3 n 1, 2, dsz m 2 z e 1 e 3 ; g 2, g 3 e 2 e 1 e 3 ; Ω 1, Ω 2, Ω 3 Ω 1 g 2, g 3, Ω 1 g 2, g 3 Ω 3 g 2, g 3, Ω 3 g 2, g 3 m Λ Ω 3 Ω 1 e n Ω n ; g 2, g 3 n 1, 2, 3 Involving theta functions ϑ 3, qm m 1 m 14 2 Km Π z Π z ϑ 1, qm 2 Km
27 Π 2 Km ϑ dz m ϑ s z m ϑ ϑ 1 0, qm 3 ϑ 3 0, qm ϑ 1 z ϑ 3 0,qm 2, qm z ϑ 3 0,qm 2, qm Zeros ds2 r 1 Km 2 s 1 K1 m m 0 ; r, s History C. G. J. Jacobi (1827) N. H. Abel (1827) J. Glaisher (1882) introduced the notation ds
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