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variants of this functions
Hypergeometric0F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1[b,z] > Transformations > Related transformations





http://functions.wolfram.com/07.17.16.0005.01









  


  










Input Form





Hypergeometric0F1[b, z] == ((2 z^((1 - b)/2))/Gamma[1 - b]) BesselK[b - 1, 2 Sqrt[z]] - ((z^(1 - b) Gamma[b - 1])/Gamma[1 - b]) Hypergeometric0F1[2 - b, z] /; !Element[b, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]]], RowBox[List["BesselK", "[", RowBox[List[RowBox[List["b", "-", "1"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]]], RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "z"]], "]"]]]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["b", ",", "Integers"]], "]"]], "]"]]]]]]










MathML Form







<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> &#8202; </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </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[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;0&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;b&quot;, Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric0F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 2 </mn> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; 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</mo> <mrow> <mi> b </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric0F1 </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> &#915; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> BesselK </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> &#915; </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> &#915; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric0F1 </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> b </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["b", "-", "1"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["b", "\[Element]", "Integers"]]]]]]]]]]










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





2001-10-29





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