html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 JacobiP

 http://functions.wolfram.com/05.06.13.0009.01

 Input Form

 z^2 Derivative[2][w][z] + z (1 - r - 2 s + (d r z^r (a - b + (2 + a + b) d z^r))/(-1 + d^2 z^(2 r))) Derivative[1][w][z] + (1/(-1 + d^2 z^(2 r))) ((-s) (r + s) - (a - b) d r s z^r - d^2 z^(2 r) (s + r n) (-s + r (1 + a + b + n))) w[z] == 0 /; w[z] == Subscript[c, 1] z^s JacobiP[n, a, b, d z^r] + Subscript[c, 2] z^s MeijerG[{{1 + n, -a - b - n}, {}}, {{0, -a}, {}}, (1 - d z^r)/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "r", "-", RowBox[List["2", " ", "s"]], "+", FractionBox[RowBox[List["d", " ", "r", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "+", "b"]], ")"]], " ", "d", " ", SuperscriptBox["z", "r"]]]]], ")"]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "s"]], " ", RowBox[List["(", RowBox[List["r", "+", "s"]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "d", " ", "r", " ", "s", " ", SuperscriptBox["z", "r"]]], "-", RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]], " ", RowBox[List["(", RowBox[List["s", "+", RowBox[List["r", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "s"]], "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", RowBox[List["d", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", "s"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", RowBox[List["d", " ", SuperscriptBox["z", "r"]]]]], "2"]]], "]"]]]]]]]]]]]]

 MathML Form

 z 2 w ′′ ( z ) + ( d r ( ( a + b + 2 ) d z r + a - b ) z r d 2 z 2 r - 1 - r - 2 s + 1 ) z w ( z ) + - ( a - b ) d r s z r - d 2 ( s + r n ) ( r ( a + b + n + 1 ) - s ) z 2 r - s ( r + s ) d 2 z 2 r - 1 w ( z ) 0 /; w ( z ) c 1 z s P n ( a , b ) ( d z r ) + c 2 z s G 2 , 2 2 , 2 ( 1 - d z r 2 n + 1 , - a - b - n 0 , - a ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", RowBox[List["d", " ", SuperscriptBox["z", "r"]]]]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["n", "+", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "n"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", "a"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] Condition z 2 z 2 w z d r a b 2 d z r a -1 b z r d 2 z 2 r -1 -1 -1 r -1 2 s 1 z z w z -1 a -1 b d r s z r -1 d 2 s r n r a b n 1 -1 s z 2 r -1 s r s d 2 z 2 r -1 -1 w z 0 w z Subscript c 1 z s JacobiP n a b d z r Subscript c 2 z s MeijerG n 1 -1 a -1 b -1 n 0 -1 a 1 -1 d z r 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List["z_", " ", RowBox[List["(", RowBox[List["1", "-", "r_", "-", RowBox[List["2", " ", "s_"]], "+", FractionBox[RowBox[List["d_", " ", "r_", " ", SuperscriptBox["z_", "r_"], " ", RowBox[List["(", RowBox[List["a_", "-", "b_", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a_", "+", "b_"]], ")"]], " ", "d_", " ", SuperscriptBox["z_", "r_"]]]]], ")"]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["d_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "s_"]], " ", RowBox[List["(", RowBox[List["r_", "+", "s_"]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a_", "-", "b_"]], ")"]], " ", "d_", " ", "r_", " ", "s_", " ", SuperscriptBox["z_", "r_"]]], "-", RowBox[List[SuperscriptBox["d_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]], " ", RowBox[List["(", RowBox[List["s_", "+", RowBox[List["r_", " ", "n_"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "s_"]], "+", RowBox[List["r_", " ", RowBox[List["(", RowBox[List["1", "+", "a_", "+", "b_", "+", "n_"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["d_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", RowBox[List["d", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["d", " ", SuperscriptBox["z", "r"]]]]], ")"]]]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02