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/07.15.06.0035.01

 Input Form

 JacobiP[\[Nu], a, b, z] == ((Sin[Pi (b + \[Nu])] (z - 1)^\[Nu])/(2^\[Nu] ((-1 - a - b - 2 \[Nu])! Sin[Pi (a + \[Nu])] Gamma[1 + \[Nu]] Gamma[1 + a + b + \[Nu]]))) Log[(z - 1)/2] Hypergeometric2F1[-\[Nu], -a - \[Nu], -a - b - 2 \[Nu], 2/(1 - z)] - ((2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] Gamma[1 + a + \[Nu]])/ Pi) (z - 1)^(-1 - a - b - \[Nu]) Sum[((Pochhammer[1 + a + b + \[Nu], k] Gamma[-1 - a - b - k - 2 \[Nu]])/ (k! Gamma[-b - k - \[Nu]])) (2/(1 - z))^k, {k, 0, -2 - a - b - 2 \[Nu]}] + (Sin[Pi (b + \[Nu])]/(2^\[Nu] (Sin[Pi (a + \[Nu])] Gamma[a + b + \[Nu] + 1] Gamma[1 + \[Nu]]))) (z - 1)^\[Nu] Sum[((Pochhammer[-a - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! (k - a - b - 2 \[Nu] - 1)!)) (PolyGamma[k + 1] + PolyGamma[k - a - b - 2 \[Nu]] - PolyGamma[k - \[Nu]] - PolyGamma[1 + a + \[Nu] - k]) (2/(1 - z))^k, {k, 0, Infinity}] /; Element[-1 - a - b - 2 \[Nu], Integers] && -1 - a - b - 2 \[Nu] > 0 && !Element[b + \[Nu], Integers] && Abs[1 - z] > 2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]]]], "\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "k", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "k", "-", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]]]], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]", "-", "k"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ">", "0"]], "&&", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], ">", "2"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) 2 - ν sin ( π ( b + ν ) ) ( z - 1 ) ν ( - a - b - 2 ν - 1 ) ! sin ( π ( a + ν ) ) Γ ( ν + 1 ) Γ ( a + b + ν + 1 ) log ( z - 1 2 ) 2 F 1 ( - ν , - a - ν ; - a - b - 2 ν ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 a + b + ν + 1 sin ( π ν ) Γ ( a + ν + 1 ) π ( z - 1 ) - a - b - ν - 1 k = 0 - a - b - 2 ν - 2 ( a + b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] Γ ( - a - b - 2 ν - k - 1 ) k ! Γ ( - b - ν - k ) ( 2 1 - z ) k + 2 - ν sin ( π ( b + ν ) ) sin ( π ( a + ν ) ) Γ ( a + b + ν + 1 ) Γ ( ν + 1 ) ( z - 1 ) ν k = 0 ( - a - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( k - a - b - 2 ν - 1 ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a - k + ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - ν ) + ψ TagBox["\[Psi]", PolyGamma] ( k - a - b - 2 ν ) ) ( 2 1 - z ) k /; - a - b - 2 ν - 1 + b + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" > 2 Condition JacobiP ν a b z 2 -1 ν b ν z -1 ν -1 a -1 b -1 2 ν -1 a ν Gamma ν 1 Gamma a b ν 1 -1 z -1 2 -1 Hypergeometric2F1 -1 ν -1 a -1 ν -1 a -1 b -1 2 ν 2 1 -1 z -1 -1 2 a b ν 1 ν Gamma a ν 1 -1 z -1 -1 a -1 b -1 ν -1 k 0 -1 a -1 b -1 2 ν -2 Pochhammer a b ν 1 k Gamma -1 a -1 b -1 2 ν -1 k -1 k Gamma -1 b -1 ν -1 k -1 2 1 -1 z -1 k 2 -1 ν b ν a ν Gamma a b ν 1 Gamma ν 1 -1 z -1 ν k 0 DirectedInfinity 1 Pochhammer -1 a -1 ν k Pochhammer -1 ν k k k -1 a -1 b -1 2 ν -1 -1 PolyGamma k 1 -1 PolyGamma a -1 k ν 1 -1 PolyGamma k -1 ν PolyGamma k -1 a -1 b -1 2 ν 2 1 -1 z -1 k -1 a -1 b -1 2 ν -1 SuperPlus b ν 1 -1 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "k", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "k", "-", "\[Nu]"]], "]"]]]]]]]]], "\[Pi]"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "+", "\[Nu]", "-", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]], "-", "1"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ">", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], ">", "2"]]]]]]]]]]

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

 2001-10-29