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 MathieuCPrime

 http://functions.wolfram.com/11.03.06.0003.01

 Input Form

 MathieuCPrime[MathieuCharacteristicA[r, q], q, z] \[Proportional] -(r Sin[r z]) + (((2 - r) Sin[(-2 + r) z])/(4 (-1 + r)) + ((2 + r) Sin[(2 + r) z])/(4 (1 + r))) q + (((4 - r) Sin[(-4 + r) z])/(32 (-2 + r) (-1 + r)) - (1/2) r (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) Sin[r z] - ((4 + r) Sin[(4 + r) z])/(32 (1 + r) (2 + r))) q^2 + (((6 - r) Sin[(-6 + r) z])/(384 (-3 + r) (-2 + r) (-1 + r)) - ((7 - 4 r + r^2) Sin[(-2 + r) z])/(128 (-1 + r)^3 (1 + r)) + ((7 + 4 r + r^2) Sin[(2 + r) z])/(128 (-1 + r) (1 + r)^3) + (1/2) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (((2 - r) Sin[(-2 + r) z])/(4 (-1 + r)) + ((2 + r) Sin[(2 + r) z])/ (4 (1 + r))) + ((6 + r) Sin[(6 + r) z])/(384 (1 + r) (2 + r) (3 + r))) q^3 + (((8 - r) Sin[(-8 + r) z])/(6144 (-4 + r) (-3 + r) (-2 + r) (-1 + r)) + ((4 - r) (10 - 5 r + r^2) Sin[(-4 + r) z])/ (768 (-3 + r) (-2 + r) (-1 + r)^3 (1 + r)) - (1/4) r ((-(3/2)) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (1/(16 (-1 + r)^2) + 1/(16 (1 + r)^2)) - 2 (1/(1024 (-2 + r)^2 (-1 + r)^2) + 1/(1024 (1 + r)^2 (2 + r)^2) + (7 - 4 r + r^2)/(256 (-2 + r) (-1 + r)^4 (1 + r)) + (7 + 4 r + r^2)/(256 (-1 + r) (1 + r)^4 (2 + r)))) Sin[r z] - ((4 + r) (10 + 5 r + r^2) Sin[(4 + r) z])/(768 (-1 + r) (1 + r)^3 (2 + r) (3 + r)) + (1/2) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (((4 - r) Sin[(-4 + r) z])/(32 (-2 + r) (-1 + r)) - ((4 + r) Sin[(4 + r) z])/(32 (1 + r) (2 + r))) - ((8 + r) Sin[(8 + r) z])/(6144 (1 + r) (2 + r) (3 + r) (4 + r))) q^4 + (((10 - r) Sin[(-10 + r) z])/(122880 (-5 + r) (-4 + r) (-3 + r) (-2 + r) (-1 + r)) + ((6 - r) (13 - 6 r + r^2) Sin[(-6 + r) z])/ (8192 (-4 + r) (-3 + r) (-2 + r) (-1 + r)^3 (1 + r)) - ((232 + 13 r + 47 r^2 - 8 r^3 + 8 r^4 - 5 r^5 + r^6) Sin[(-2 + r) z])/ (3072 (-3 + r) (-1 + r)^5 (1 + r)^3 (2 + r)) + ((232 - 13 r + 47 r^2 + 8 r^3 + 8 r^4 + 5 r^5 + r^6) Sin[(2 + r) z])/ (3072 (-2 + r) (-1 + r)^3 (1 + r)^5 (3 + r)) + (1/4) ((-(3/2)) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (1/(16 (-1 + r)^2) + 1/(16 (1 + r)^2)) - 2 (1/(1024 (-2 + r)^2 (-1 + r)^2) + 1/(1024 (1 + r)^2 (2 + r)^2) + (7 - 4 r + r^2)/(256 (-2 + r) (-1 + r)^4 (1 + r)) + (7 + 4 r + r^2)/(256 (-1 + r) (1 + r)^4 (2 + r)))) (((2 - r) Sin[(-2 + r) z])/(4 (-1 + r)) + ((2 + r) Sin[(2 + r) z])/ (4 (1 + r))) + ((6 + r) (13 + 6 r + r^2) Sin[(6 + r) z])/ (8192 (-1 + r) (1 + r)^3 (2 + r) (3 + r) (4 + r)) + (1/2) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (((6 - r) Sin[(-6 + r) z])/(384 (-3 + r) (-2 + r) (-1 + r)) - ((7 - 4 r + r^2) Sin[(-2 + r) z])/(128 (-1 + r)^3 (1 + r)) + ((7 + 4 r + r^2) Sin[(2 + r) z])/(128 (-1 + r) (1 + r)^3) + ((6 + r) Sin[(6 + r) z])/(384 (1 + r) (2 + r) (3 + r))) + ((10 + r) Sin[(10 + r) z])/(122880 (1 + r) (2 + r) (3 + r) (4 + r) (5 + r))) q^5 + (((12 - r) Sin[(-12 + r) z])/ (2949120 (-6 + r) (-5 + r) (-4 + r) (-3 + r) (-2 + r) (-1 + r)) + ((8 - r) (16 - 7 r + r^2) Sin[(-8 + r) z])/(122880 (-5 + r) (-4 + r) (-3 + r) (-2 + r) (-1 + r)^3 (1 + r)) - ((19268 - 15344 r + 4739 r^2 - 2938 r^3 + 1945 r^4 - 1180 r^5 + 533 r^6 - 122 r^7 + 11 r^8) Sin[(-4 + r) z])/ (196608 (-3 + r) (-2 + r)^3 (-1 + r)^5 (1 + r)^3 (2 + r)) - (1/6) r (-2 (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (1/(1024 (-2 + r)^2 (-1 + r)^2) + 1/(1024 (1 + r)^2 (2 + r)^2) + (7 - 4 r + r^2)/(256 (-2 + r) (-1 + r)^4 (1 + r)) + (7 + 4 r + r^2)/(256 (-1 + r) (1 + r)^4 (2 + r))) - 3 (1/(147456 (-3 + r)^2 (-2 + r)^2 (-1 + r)^2) + 1/(147456 (1 + r)^2 (2 + r)^2 (3 + r)^2) + (10 - 5 r + r^2)/ (12288 (-3 + r) (-2 + r)^2 (-1 + r)^4 (1 + r)) + (7 - 4 r + r^2)^2/(16384 (-2 + r)^2 (-1 + r)^6 (1 + r)^2) + (7 + 4 r + r^2)^2/(16384 (-1 + r)^2 (1 + r)^6 (2 + r)^2) + (10 + 5 r + r^2)/(12288 (-1 + r) (1 + r)^4 (2 + r)^2 (3 + r)) + (232 + 13 r + 47 r^2 - 8 r^3 + 8 r^4 - 5 r^5 + r^6)/ (6144 (-3 + r) (-2 + r) (-1 + r)^6 (1 + r)^3 (2 + r)) + (232 - 13 r + 47 r^2 + 8 r^3 + 8 r^4 + 5 r^5 + r^6)/ (6144 (-2 + r) (-1 + r)^3 (1 + r)^6 (2 + r) (3 + r))) - (5/4) (1/(16 (-1 + r)^2) + 1/(16 (1 + r)^2)) ((-(3/2)) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (1/(16 (-1 + r)^2) + 1/(16 (1 + r)^2)) - 2 (1/(1024 (-2 + r)^2 (-1 + r)^2) + 1/(1024 (1 + r)^2 (2 + r)^2) + (7 - 4 r + r^2)/(256 (-2 + r) (-1 + r)^4 (1 + r)) + (7 + 4 r + r^2)/(256 (-1 + r) (1 + r)^4 (2 + r))))) Sin[r z] - ((19268 + 15344 r + 4739 r^2 + 2938 r^3 + 1945 r^4 + 1180 r^5 + 533 r^6 + 122 r^7 + 11 r^8) Sin[(4 + r) z])/ (196608 (-2 + r) (-1 + r)^3 (1 + r)^5 (2 + r)^3 (3 + r)) + (1/4) ((-(3/2)) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (1/(16 (-1 + r)^2) + 1/(16 (1 + r)^2)) - 2 (1/(1024 (-2 + r)^2 (-1 + r)^2) + 1/(1024 (1 + r)^2 (2 + r)^2) + (7 - 4 r + r^2)/(256 (-2 + r) (-1 + r)^4 (1 + r)) + (7 + 4 r + r^2)/(256 (-1 + r) (1 + r)^4 (2 + r)))) (((4 - r) Sin[(-4 + r) z])/(32 (-2 + r) (-1 + r)) - ((4 + r) Sin[(4 + r) z])/(32 (1 + r) (2 + r))) - ((8 + r) (16 + 7 r + r^2) Sin[(8 + r) z])/(122880 (-1 + r) (1 + r)^3 (2 + r) (3 + r) (4 + r) (5 + r)) + (1/2) (-(1/(16 (-1 + r)^2)) - 1/(16 (1 + r)^2)) (((8 - r) Sin[(-8 + r) z])/(6144 (-4 + r) (-3 + r) (-2 + r) (-1 + r)) + ((4 - r) (10 - 5 r + r^2) Sin[(-4 + r) z])/(768 (-3 + r) (-2 + r) (-1 + r)^3 (1 + r)) - ((4 + r) (10 + 5 r + r^2) Sin[(4 + r) z])/ (768 (-1 + r) (1 + r)^3 (2 + r) (3 + r)) - ((8 + r) Sin[(8 + r) z])/ (6144 (1 + r) (2 + r) (3 + r) (4 + r))) - ((12 + r) Sin[(12 + r) z])/ (2949120 (1 + r) (2 + r) (3 + r) (4 + r) (5 + r) (6 + r))) q^6 + SeriesData[\$CellContext`q, 0, {}, 0, 7, 1] /; !(Element[r, Integers] && -6 <= r <= 6)

 Standard Form

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 MathML Form

 Ce ( a MathieuCharacteristicA r ( q ) , q , z ) - r sin ( r z ) + 1 4 ( ( 2 - r ) sin ( ( r - 2 ) z ) r - 1 + ( r + 2 ) sin ( ( r + 2 ) z ) r + 1 ) q + 1 32 ( ( 4 - r ) sin ( ( r - 4 ) z ) ( r - 2 ) ( r - 1 ) + 2 r ( r 2 + 1 ) sin ( r z ) ( r - 1 ) 2 ( r + 1 ) 2 - ( r + 4 ) sin ( ( r + 4 ) z ) ( r + 1 ) ( r + 2 ) ) q 2 + 1 384 ( - ( r - 6 ) sin ( ( r - 6 ) z ) ( r - 3 ) ( r - 2 ) ( r - 1 ) + 3 ( r 3 - r 2 - r - 11 ) sin ( ( r - 2 ) z ) ( r - 1 ) 3 ( r + 1 ) 2 - 3 ( r 3 + r 2 - r + 11 ) sin ( ( r + 2 ) z ) ( r - 1 ) 2 ( r + 1 ) 3 + ( r + 6 ) sin ( ( r + 6 ) z ) ( r + 1 ) ( r + 2 ) ( r + 3 ) ) q 3 + 1 6144 ( - ( r - 8 ) sin ( ( r - 8 ) z ) ( r - 4 ) ( r - 3 ) ( r - 2 ) ( r - 1 ) + 4 ( r - 4 ) ( r 3 - r 2 - 7 r - 29 ) sin ( ( r - 4 ) z ) ( r - 3 ) ( r - 2 ) ( r - 1 ) 3 ( r + 1 ) 2 - 6 r ( r 8 - 15 r 6 - 185 r 4 + 675 r 2 + 316 ) sin ( r z ) ( r - 2 ) 2 ( r - 1 ) 4 ( r + 1 ) 4 ( r + 2 ) 2 + 4 ( r + 4 ) ( r 3 + r 2 - 7 r + 29 ) sin ( ( r + 4 ) z ) ( r - 1 ) 2 ( r + 1 ) 3 ( r + 2 ) ( r + 3 ) - ( r + 8 ) sin ( ( r + 8 ) z ) ( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + 4 ) ) q 4 + 1 122880 ( - ( r - 10 ) sin ( ( r - 10 ) z ) ( r - 5 ) ( r - 4 ) ( r - 3 ) ( r - 2 ) ( r - 1 ) + 5 ( r - 6 ) ( r 3 - r 2 - 17 r - 55 ) sin ( ( r - 6 ) z ) ( r - 4 ) ( r - 3 ) ( r - 2 ) ( r - 1 ) 3 ( r + 1 ) 2 - ( 10 ( r 9 - r 8 - 31 r 7 - 5 r 6 - 273 r 5 + 2457 r 4 + 1931 r 3 - 6335 r 2 - 3572 r - 7564 ) sin ( ( r - 2 ) z ) ) / ( ( r - 3 ) ( r - 2 ) ( r - 1 ) 5 ( r + 1 ) 4 ( r + 2 ) 2 ) + ( 10 ( r 9 + r 8 - 31 r 7 + 5 r 6 - 273 r 5 - 2457 r 4 + 1931 r 3 + 6335 r 2 - 3572 r + 7564 ) sin ( ( r + 2 ) z ) ) / ( ( r - 2 ) 2 ( r - 1 ) 4 ( r + 1 ) 5 ( r + 2 ) ( r + 3 ) ) - 5 ( r + 6 ) ( r 3 + r 2 - 17 r + 55 ) sin ( ( r + 6 ) z ) ( r - 1 ) 2 ( r + 1 ) 3 ( r + 2 ) ( r + 3 ) ( r + 4 ) + ( r + 10 ) sin ( ( r + 10 ) z ) ( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + 4 ) ( r + 5 ) ) q 5 + 1 2949120 ( - ( r - 12 ) sin ( ( r - 12 ) z ) ( r - 6 ) ( r - 5 ) ( r - 4 ) ( r - 3 ) ( r - 2 ) ( r - 1 ) + 6 ( r - 8 ) ( r 3 - r 2 - 31 r - 89 ) sin ( ( r - 8 ) z ) ( r - 5 ) ( r - 4 ) ( r - 3 ) ( r - 2 ) ( r - 1 ) 3 ( r + 1 ) 2 - ( 15 ( r 10 - 3 r 9 - 53 r 8 + 69 r 7 + 145 r 6 + 8211 r 5 - 16879 r 4 - 32025 r 3 + 32954 r 2 + 16404 r + 71528 ) sin ( ( r - 4 ) z ) ) / ( ( r - 3 ) ( r - 2 ) 3 ( r - 1 ) 5 ( r + 1 ) 4 ( r + 2 ) 2 ) + ( 20 r ( r 14 - 72 r 12 + 597 r 10 + 75244 r 8 - 718317 r 6 + 153312 r 4 + 4883287 r 2 + 1329084 ) sin ( r z ) ) / ( ( r - 3 ) 2 ( r - 2 ) 2 ( r - 1 ) 6 ( r + 1 ) 6 ( r + 2 ) 2 ( r + 3 ) 2 ) - ( 15 ( r 10 + 3 r 9 - 53 r 8 - 69 r 7 + 145 r 6 - 8211 r 5 - 16879 r 4 + 32025 r 3 + 32954 r 2 - 16404 r + 71528 ) sin ( ( r + 4 ) z ) ) / ( ( r - 2 ) 2 ( r - 1 ) 4 ( r + 1 ) 5 ( r + 2 ) 3 ( r + 3 ) ) + 6 ( r + 8 ) ( r 3 + r 2 - 31 r + 89 ) sin ( ( r + 8 ) z ) ( r - 1 ) 2 ( r + 1 ) 3 ( r + 2 ) ( r + 3 ) ( r + 4 ) ( r + 5 ) - ( r + 12 ) sin ( ( r + 12 ) z ) ( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + 4 ) ( r + 5 ) ( r + 6 ) ) q 6 + O ( q 7 ) SeriesData \$CellContext`q 0 -1 \$CellContext`r \$CellContext`r \$CellContext`z -1 4 -2 \$CellContext`r -1 \$CellContext`r -1 -2 \$CellContext`r \$CellContext`z 1 4 1 \$CellContext`r -1 2 \$CellContext`r 2 \$CellContext`r \$CellContext`z -1 32 -4 \$CellContext`r -2 \$CellContext`r -1 -1 \$CellContext`r -1 -4 \$CellContext`r \$CellContext`z -1 2 \$CellContext`r -1 16 -1 \$CellContext`r -2 -1 16 1 \$CellContext`r -2 \$CellContext`r \$CellContext`z -1 32 1 \$CellContext`r -1 2 \$CellContext`r -1 4 \$CellContext`r 4 \$CellContext`r \$CellContext`z -1 384 -6 \$CellContext`r -3 \$CellContext`r -1 -2 \$CellContext`r -1 -1 \$CellContext`r -1 -6 \$CellContext`r \$CellContext`z -1 128 -1 \$CellContext`r -3 1 \$CellContext`r -1 7 -1 4 \$CellContext`r \$CellContext`r 2 -2 \$CellContext`r \$CellContext`z 1 128 -1 \$CellContext`r -1 1 \$CellContext`r -3 7 4 \$CellContext`r \$CellContext`r 2 2 \$CellContext`r \$CellContext`z 1 2 -1 16 -1 \$CellContext`r -2 -1 16 1 \$CellContext`r -2 -1 4 -2 \$CellContext`r -1 \$CellContext`r -1 -2 \$CellContext`r \$CellContext`z 1 4 1 \$CellContext`r -1 2 \$CellContext`r 2 \$CellContext`r \$CellContext`z 1 384 1 \$CellContext`r -1 2 \$CellContext`r -1 3 \$CellContext`r -1 6 \$CellContext`r 6 \$CellContext`r \$CellContext`z -1 6144 -8 \$CellContext`r -4 \$CellContext`r -1 -3 \$CellContext`r -1 -2 \$CellContext`r -1 -1 \$CellContext`r -1 -8 \$CellContext`r \$CellContext`z -1 768 -4 \$CellContext`r -3 \$CellContext`r -1 -2 \$CellContext`r -1 -1 \$CellContext`r -3 1 \$CellContext`r -1 10 -1 5 \$CellContext`r \$CellContext`r 2 -4 \$CellContext`r \$CellContext`z -1 4 \$CellContext`r -3 2 -1 16 -1 \$CellContext`r -2 -1 16 1 \$CellContext`r -2 1 16 -1 \$CellContext`r -2 1 16 1 \$CellContext`r -2 -1 2 1 1024 -2 \$CellContext`r -2 -1 \$CellContext`r -2 1 1024 1 \$CellContext`r -2 2 \$CellContext`r -2 1 256 -2 \$CellContext`r -1 -1 \$CellContext`r -4 1 \$CellContext`r -1 7 -1 4 \$CellContext`r \$CellContext`r 2 1 256 -1 \$CellContext`r -1 1 \$CellContext`r -4 2 \$CellContext`r -1 7 4 \$CellContext`r \$CellContext`r 2 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29