Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Hypergeometric0F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1[b,z] > Integration > Indefinite integration > Involving functions of the direct function and elementary functions > Involving elementary functions of the direct function and elementary functions > Involving products of the direct function and a power function





http://functions.wolfram.com/07.17.21.0005.01









  


  








Input Form





Integrate[z^((b + c - 3)/2) Hypergeometric0F1[b, z] Hypergeometric0F1[c, z], z] == (2/(b + c - 1)) z^((b + c - 1)/2) HypergeometricPFQ[ {(b + c - 1)/2, (b + c - 1)/2, (b + c)/2}, {b, c, b + c - 1, (b + c + 1)/2}, 4 z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z", FractionBox[RowBox[List["b", "+", "c", "-", "3"]], "2"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["c", ",", "z"]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["2", RowBox[List["b", "+", "c", "-", "1"]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["b", "+", "c", "-", "1"]], "2"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["b", "+", "c", "-", "1"]], "2"], ",", FractionBox[RowBox[List["b", "+", "c", "-", "1"]], "2"], ",", FractionBox[RowBox[List["b", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "c", ",", RowBox[List["b", "+", "c", "-", "1"]], ",", FractionBox[RowBox[List["b", "+", "c", "+", "1"]], "2"]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <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> &#8290; </mo> <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> c </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;c&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> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 4 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;b&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;c&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;, &quot;-&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;z&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric0F1 </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <ci> Hypergeometric0F1 </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <ci> b </ci> <ci> c </ci> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["b_", "+", "c_", "-", "3"]], ")"]]]]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["c_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["b", "+", "c", "-", "1"]], ")"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["b", "+", "c", "-", "1"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["b", "+", "c", "-", "1"]], ")"]]]], ",", FractionBox[RowBox[List["b", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", "c", ",", RowBox[List["b", "+", "c", "-", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", "1"]], ")"]]]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], RowBox[List["b", "+", "c", "-", "1"]]]]]]]


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





2001-10-29