|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/03.09.21.0012.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[z^(1 - \[Mu] - \[Nu]) StruveH[\[Mu], z] StruveH[\[Nu], z], z] ==
(-(1/(2 (-1 + \[Mu] + \[Nu]))))
(-((2^(1 - \[Mu] - \[Nu]) z^2 (HypergeometricPFQ[{1, 1},
{3/2, 2, 1/2 + \[Mu]}, -(z^2/4)] + HypergeometricPFQ[{1, 1},
{3/2, 2, 1/2 + \[Nu]}, -(z^2/4)]))/(Pi Gamma[1/2 + \[Mu]]
Gamma[1/2 + \[Nu]])) + z^(2 - \[Mu] - \[Nu])
(StruveH[-1 + \[Mu], z] StruveH[-1 + \[Nu], z] +
StruveH[\[Mu], z] StruveH[\[Nu], z]))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> μ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mtext> </mtext> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> + </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> StruveH </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> StruveH </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> StruveH </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> StruveH </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> StruveH </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> StruveH </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> μ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["1", "-", "\[Mu]_", "-", "\[Nu]_"]]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|