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http://functions.wolfram.com/01.01.06.0021.01
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Sqrt[z] == Sqrt[x] Exp[Pi I Floor[Arg[z - x]/(2 Pi)]]
HypergeometricPFQ[{-(1/2)}, {}, -((z - x)/x)] /; Element[x, Reals] && x < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sqrt", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["Sqrt", "[", "x", "]"]], RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["z", "-", "x"]], "x"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "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> <msqrt> <mi> z </mi> </msqrt> <mo> ⩵ </mo> <mrow> <msqrt> <mi> x </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mo>   </mo> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mi> x </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["0", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["z", "-", "x"]], "x"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox["z_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox["x"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["z", "-", "x"]], "x"]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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