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 LegendreP

 http://functions.wolfram.com/05.07.21.0004.01

 Input Form

 Integrate[LegendreP[n, m, x], {x, 0, 1}] == (((-1)^m Pi (m + n)! 2^(-1 - 2 m))/(Gamma[(m + 1)/2] Gamma[(m + 3)/2] (n - m)!)) HypergeometricPFQ[{(1/2) (m + n + 1), (m - n)/2, m/2 + 1}, {m + 1, (m + 3)/2}, 1] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", "x"]], "]"]], RowBox[List["\[DifferentialD]", "x"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", "\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n"]], ")"]], "!"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "m"]]]]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["m", "+", "1"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["m", "+", "3"]], "2"], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]]]], ",", FractionBox[RowBox[List["m", "-", "n"]], "2"], ",", RowBox[List[FractionBox["m", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["m", "+", "3"]], "2"]]], "}"]], ",", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 0 1 P TagBox["P", LegendreP] n m ( x ) x ( - 1 ) m π ( m + n ) ! 2 - 2 m - 1 Γ ( m + 1 2 ) Γ ( m + 3 2 ) ( n - m ) ! 3 F 2 ( 1 2 ( m + n + 1 ) , m - n 2 , m 2 + 1 ; m + 1 , m + 3 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["m", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["m", "+", "3"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; n m Condition x 0 1 LegendreP n m x -1 m m n 2 -2 m -1 Gamma m 1 2 -1 Gamma m 3 2 -1 n -1 m -1 HypergeometricPFQ 1 2 m n 1 m -1 n 2 -1 m 2 -1 1 m 1 m 3 2 -1 1 n m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "x_"]], "]"]], RowBox[List["\[DifferentialD]", "x_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", "\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n"]], ")"]], "!"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "m"]]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]]]], ",", FractionBox[RowBox[List["m", "-", "n"]], "2"], ",", RowBox[List[FractionBox["m", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["m", "+", "3"]], "2"]]], "}"]], ",", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["m", "+", "1"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["m", "+", "3"]], "2"], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18