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http://functions.wolfram.com/07.18.07.0008.01
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Hypergeometric0F1Regularized[b, z] ==
(1/(2 Pi I)) Integrate[E^(s + z/s)/s^b, {s, \[Gamma] - I Infinity,
\[Gamma] + I Infinity}] /; \[Gamma] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["1", " "]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["s", "+", FractionBox["z", "s"]]]], " ", SuperscriptBox["s", RowBox[List["-", "b"]]]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", " ", RowBox[List["\[Gamma]", ">", "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> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mi> b </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> <mo> + </mo> <mi> γ </mi> </mrow> </msubsup> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> z </mi> <mi> s </mi> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> s </mi> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> γ </mi> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric0F1Regularized </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> s </ci> </bvar> <lowlimit> <apply> <plus /> <ci> γ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <imaginaryi /> <infinity /> </apply> <ci> γ </ci> </apply> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> s </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["s", "+", FractionBox["z", "s"]]]], " ", SuperscriptBox["s", RowBox[List["-", "b"]]]]], RowBox[List["\[DifferentialD]", "s"]]]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List["\[Gamma]", ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQRegularized[{a},{b},z] | HypergeometricPFQRegularized[{a1,a2},{b1},z] | HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] | |
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