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http://functions.wolfram.com/12.06.09.0001.01
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Sign[x] == Limit[((4 n! Gamma[m + 3/2] Gamma[m + n + 2])/
(Sqrt[Pi] m! Gamma[n + 1/2] Gamma[m + n + 3/2])) x
(HypergeometricPFQ[{-m, 1/2 - n, m + n + 2}, {3/2, 3/2}, x^2]/
HypergeometricPFQ[{-n, -m - 1/2, m + n + 3/2}, {1/2, 1}, x^2]),
n + m -> Infinity] /; -1 < x < 1 && Element[n, Integers] && n >= 0 &&
Element[m, Integers] && m >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sign", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["4", " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", FractionBox["3", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "n", "+", "2"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "n", "+", FractionBox["3", "2"]]], "]"]], " "]]], " ", "x", " ", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List[FractionBox["1", "2"], "-", "n"]], ",", RowBox[List["m", "+", "n", "+", "2"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["x", "2"]]], "]"]], "/", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "m"]], "-", FractionBox["1", "2"]]], ",", RowBox[List["m", "+", "n", "+", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", SuperscriptBox["x", "2"]]], "]"]]]]]], ",", RowBox[List[RowBox[List["n", "+", "m"]], "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mfrac> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mfrac> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["-", "m"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["x", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "m"]], "-", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["x", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Sign </ci> <ci> x </ci> </apply> <apply> <limit /> <bvar> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </bvar> <condition> <apply> <tendsto /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <infinity /> </apply> </condition> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> <apply> <times /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='integer'> 1 </cn> </list> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", FractionBox["3", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "n", "+", "2"]], "]"]]]], ")"]], " ", "x", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List[FractionBox["1", "2"], "-", "n"]], ",", RowBox[List["m", "+", "n", "+", "2"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["x", "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "n", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "m"]], "-", FractionBox["1", "2"]]], ",", RowBox[List["m", "+", "n", "+", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", SuperscriptBox["x", "2"]]], "]"]]]]], ",", RowBox[List[RowBox[List["n", "+", "m"]], "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
