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JacobiSD






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSD[z,m] > Series representations > Generalized power series > Expansions at m==1





http://functions.wolfram.com/09.35.06.0015.01









  


  










Input Form





JacobiSD[z, m] \[Proportional] Sinh[z] + (1/16) (4 z Cosh[z] - 7 Sinh[z] + Sinh[3 z]) (m - 1) + (1/256) (-48 z Cosh[z] + 12 z Cosh[3 z] + 2 (32 + 4 z^2 - 15 Cosh[2 z] + Cosh[4 z]) Sinh[z]) (m - 1)^2 + (1/12288) (8 z (237 + 4 z^2) Cosh[z] - 756 z Cosh[3 z] + 60 z Cosh[5 z] - 3 (1013 + 136 z^2) Sinh[z] + 6 (121 + 36 z^2) Sinh[3 z] - 72 Sinh[5 z] + 3 Sinh[7 z]) (m - 1)^3 + (1/196608) (-4 z (6555 + 176 z^2) Cosh[z] + 216 z (61 + 4 z^2) Cosh[3 z] - 1740 z Cosh[5 z] + 84 z Cosh[7 z] + 2 (20679 + 3144 z^2 + 16 z^4) Sinh[z] - 3 (3647 + 1872 z^2) Sinh[3 z] + 6 (227 + 100 z^2) Sinh[5 z] - 96 Sinh[7 z] + 3 Sinh[9 z]) (m - 1)^4 + (1/15728640) (8 z (233445 + 7820 z^2 + 16 z^4) Cosh[z] - 1080 z (1005 + 124 z^2) Cosh[3 z] + 200 z (927 + 100 z^2) Cosh[5 z] - 15540 z Cosh[7 z] + 540 z Cosh[9 z] - 360 (8091 + 1324 z^2 + 12 z^4) Sinh[z] + 45 (18391 + 12360 z^2 + 288 z^4) Sinh[3 z] - 1200 (99 + 85 z^2) Sinh[5 z] + 30 (365 + 196 z^2) Sinh[7 z] - 600 Sinh[9 z] + 15 Sinh[11 z]) (m - 1)^5 + (1/754974720) (-12 z (6783615 + 259960 z^2 + 1024 z^4) Cosh[z] + 432 z (121345 + 20670 z^2 + 216 z^4) Cosh[3 z] - 4500 z (2343 + 520 z^2) Cosh[5 z] + 840 z (1401 + 196 z^2) Cosh[7 z] - 72900 z Cosh[9 z] + 1980 z Cosh[11 z] + 8 (15734205 + 2706615 z^2 + 32700 z^4 + 32 z^6) Sinh[z] - 270 (139787 + 111036 z^2 + 5184 z^4) Sinh[3 z] + 15 (399333 + 484200 z^2 + 20000 z^4) Sinh[5 z] - 5040 (130 + 147 z^2) Sinh[7 z] + 90 (535 + 324 z^2) Sinh[9 z] - 2160 Sinh[11 z] + 45 Sinh[13 z]) (m - 1)^6 + (1/84557168640) (4 z (2099102355 + 88220160 z^2 + 486528 z^4 + 256 z^6) Cosh[z] - 756 z (7728505 + 1612920 z^2 + 35424 z^4) Cosh[3 z] + 2100 z (628377 + 204200 z^2 + 4000 z^4) Cosh[5 z] - 2940 z (60261 + 18424 z^2) Cosh[7 z] + 204120 z (73 + 12 z^2) Cosh[9 z] - 734580 z Cosh[11 z] + 16380 z Cosh[13 z] - 14 (920985345 + 164375460 z^2 + 2373840 z^4 + 4736 z^6) Sinh[z] + 63 (64038815 + 57130200 z^2 + 3836160 z^4 + 20736 z^6) Sinh[3 z] - 105 (6578709 + 9930000 z^2 + 880000 z^4) Sinh[5 z] + 105 (812787 + 1367688 z^2 + 76832 z^4) Sinh[7 z] - 25200 (301 + 405 z^2) Sinh[9 z] + 6930 (67 + 44 z^2) Sinh[11 z] - 17640 Sinh[13 z] + 315 Sinh[15 z]) (m - 1)^7 + (1/1352914698240) (-84 z (1488442935 + 66969680 z^2 + 456736 z^4 + 512 z^6) Cosh[z] + 432 z (214135145 + 51519300 z^2 + 1681344 z^4 + 5184 z^6) Cosh[3 z] - 42000 z (543171 + 225500 z^2 + 9800 z^4) Cosh[5 z] + 588 z (5945595 + 2777320 z^2 + 76832 z^4) Cosh[7 z] - 34020 z (10553 + 3960 z^2) Cosh[9 z] + 9240 z (2637 + 484 z^2) Cosh[11 z] - 999180 z Cosh[13 z] + 18900 z Cosh[15 z] + (190895208525 + 35062219080 z^2 + 573071520 z^4 + 1662976 z^6 + 512 z^8) Sinh[z] - 63 (981794585 + 955001880 z^2 + 80866080 z^4 + 953856 z^6) Sinh[3 z] + 70 (160660881 + 282908700 z^2 + 37800000 z^4 + 400000 z^6) Sinh[5 z] - 1680 (907212 + 1973769 z^2 + 249704 z^4) Sinh[7 z] + 945 (164329 + 341928 z^2 + 23328 z^4) Sinh[9 z] - 5040 (2312 + 3509 z^2) Sinh[11 z] + 630 (971 + 676 z^2) Sinh[13 z] - 20160 Sinh[15 z] + 315 Sinh[17 z]) (m - 1)^8 + (1/194819716546560) (16 z (1055822405715 + 50087404080 z^2 + 396402552 z^4 + 666432 z^6 + 128 z^8) Cosh[z] - 1944 z (6760275865 + 1809037860 z^2 + 76241088 z^4 + 528768 z^6) Cosh[3 z] + 4500 z (772279011 + 381412920 z^2 + 25603200 z^4 + 160000 z^6) Cosh[5 z] - 15876 z (37107615 + 22857520 z^2 + 1459808 z^4) Cosh[7 z] + 20412 z (3423715 + 2027160 z^2 + 69984 z^4) Cosh[9 z] - 124740 z (46943 + 20328 z^2) Cosh[11 z] + 98280 z (3399 + 676 z^2) Cosh[13 z] - 11736900 z Cosh[15 z] + 192780 z Cosh[17 z] - 18 (1425876118065 + 268044548940 z^2 + 4803020880 z^4 + 17713920 z^6 + 12032 z^8) Sinh[z] + 162 (52863102635 + 55014259860 z^2 + 5487138720 z^4 + 98775936 z^6 + 186624 z^8) Sinh[3 z] - 1890 (865438233 + 1710777900 z^2 + 298400000 z^4 + 7200000 z^6) Sinh[5 z] + 126 (1893369285 + 4946017860 z^2 + 977303040 z^4 + 15059072 z^6) Sinh[7 z] - 68040 (396593 + 1098306 z^2 + 174960 z^4) Sinh[9 z] + 945 (2483103 + 6002568 z^2 + 468512 z^4) Sinh[11 z] - 136080 (1121 + 1859 z^2) Sinh[13 z] + 5670 (1237 + 900 z^2) Sinh[15 z] - 204120 Sinh[17 z] + 2835 Sinh[19 z]) (m - 1)^9 + (1/15585577323724800) (-80 z (15954035719545 + 789796360080 z^2 + 6985982808 z^4 + 15268032 z^6 + 6656 z^8) Cosh[z] + 4860 z (213083709055 + 61972017240 z^2 + 3132700704 z^4 + 33903360 z^6 + 41472 z^8) Cosh[3 z] - 18000 z (16109960643 + 9069298350 z^2 + 809445000 z^4 + 11800000 z^6) Cosh[5 z] + 17640 z (3019923495 + 2268001260 z^2 + 230496000 z^4 + 2151296 z^6) Cosh[7 z] - 510300 z (13834049 + 11072592 z^2 + 909792 z^4) Cosh[9 z] + 41580 z (16583175 + 11620840 z^2 + 468512 z^4) Cosh[11 z] - 245700 z (199353 + 95992 z^2) Cosh[13 z] + 5103000 z (473 + 100 z^2) Cosh[15 z] - 74220300 z Cosh[17 z] + 1077300 z Cosh[19 z] + (1930805954624925 + 370045682269200 z^2 + 7123026708000 z^4 + 31120266240 z^6 + 32509440 z^8 + 4096 z^10) Sinh[z] - 8505 (77680098395 + 85379407560 z^2 + 9631175040 z^4 + 228407040 z^6 + 995328 z^8) Sinh[3 z] + 225 (584519733567 + 1265178448800 z^2 + 266928480000 z^4 + 10152800000 z^6 + 40000000 z^8) Sinh[5 z] - 315 (64674151155 + 194009059440 z^2 + 51409827840 z^4 + 1867324928 z^6) Sinh[7 z] + 5670 (441741125 + 1502509500 z^2 + 384212160 z^4 + 7558272 z^6) Sinh[9 z] - 75600 (3227163 + 10623921 z^2 + 1991176 z^4) Sinh[11 z] + 4725 (3922749 + 10606440 z^2 + 913952 z^4) Sinh[13 z] - 1134000 (938 + 1665 z^2) Sinh[15 z] + 28350 (1535 + 1156 z^2) Sinh[17 z] - 1134000 Sinh[19 z] + 14175 Sinh[21 z]) (m - 1)^10 + O[(m - 1)^11]










Standard Form





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MathML Form







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Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02





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