Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[z] > Integration > Definite integration





http://functions.wolfram.com/03.16.21.0002.01









  


  










Input Form





Integrate[(t^(\[Alpha] - 1) KelvinKer[t])/E^(p t), {t, 0, Infinity}] == (1/3) 2^(-3 + \[Alpha]) (2 p Gamma[(1 + \[Alpha])/2]^2 (-6 Cos[(1/4) Pi (1 + \[Alpha])] HypergeometricPFQ[ {1/4 + \[Alpha]/4, 1/4 + \[Alpha]/4, 3/4 + \[Alpha]/4, 3/4 + \[Alpha]/4}, {1/2, 3/4, 5/4}, -p^4] + p^2 (1 + \[Alpha])^2 Cos[(1/4) (Pi - Pi \[Alpha])] HypergeometricPFQ[{3/4 + \[Alpha]/4, 3/4 + \[Alpha]/4, 5/4 + \[Alpha]/4, 5/4 + \[Alpha]/4}, {5/4, 3/2, 7/4}, -p^4]) + 3 Gamma[\[Alpha]/2]^2 (2 Cos[(Pi \[Alpha])/4] HypergeometricPFQ[ {1/2 + \[Alpha]/4, 1/2 + \[Alpha]/4, \[Alpha]/4, \[Alpha]/4}, {1/4, 1/2, 3/4}, -p^4] - p^2 \[Alpha]^2 Sin[(Pi \[Alpha])/4] HypergeometricPFQ[{1/2 + \[Alpha]/4, 1/2 + \[Alpha]/4, 1 + \[Alpha]/4, 1 + \[Alpha]/4}, {3/4, 5/4, 3/2}, -p^4])) /; Re[\[Alpha]] > 0 && Re[p] > -(1/Sqrt[2])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", "t"]]], RowBox[List["KelvinKer", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p", " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["p", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["\[Pi]", " ", "\[Alpha]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox["7", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["\[Alpha]", "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", FractionBox["\[Alpha]", "4"], ",", FractionBox["\[Alpha]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "2"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["p", "2"], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", RowBox[List["-", FractionBox["1", SqrtBox["2"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> ker </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> p </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;4&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;p&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> p </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;4&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;p&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 7 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> p </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;4&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;5&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;5&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;5&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;7&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;p&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#945; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> p </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;4&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Alpha]&quot;, &quot;4&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;p&quot;, &quot;4&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <ci> t </ci> </apply> </apply> <apply> <ci> KelvinKer </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <pi /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='rational'> 5 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 5 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 5 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 7 <sep /> 4 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='rational'> 5 <sep /> 4 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> &#945; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", "t_"]]], " ", RowBox[List["KelvinKer", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p", " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["p", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["\[Pi]", " ", "\[Alpha]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox["7", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["\[Alpha]", "2"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", FractionBox["\[Alpha]", "4"], ",", FractionBox["\[Alpha]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "2"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["p", "2"], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["p", "4"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", RowBox[List["-", FractionBox["1", SqrtBox["2"]]]]]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.