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http://functions.wolfram.com/08.02.26.0075.01
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UnitStep[1 - Abs[z]] Sqrt[Sqrt[-1 + z] + Sqrt[z]]
EllipticK[(2 Sqrt[-1 + z])/(Sqrt[-1 + z] - Sqrt[z])] ==
(Pi/2) MeijerG[{{}, {3/4, 3/4}}, {{0, 1/2}, {}}, z] /;
!IntervalMemberQ[Interval[{-1, 0}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], "+", SqrtBox["z"]]]], " ", " ", RowBox[List["EllipticK", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], "-", SqrtBox["z"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["3", "4"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]
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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> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z_", "]"]]]], "]"]], " ", SqrtBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z_"]]], "+", SqrtBox["z_"]]]], " ", RowBox[List["EllipticK", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z_"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z_"]]], "-", SqrtBox["z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["3", "4"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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