|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/03.21.21.0061.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[SphericalBesselJ[\[Nu], t], {t, 0, Infinity}] ==
(2^(-2 + \[Alpha]) Sqrt[Pi] Gamma[(\[Alpha] + \[Nu])/2])/
Gamma[(1/2) (3 - \[Alpha] + \[Nu])] /; Re[\[Alpha] + \[Nu]] > 0 &&
Re[\[Alpha]] < 2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "2"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> α </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 2 </mn> </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> <ci> SphericalBesselJ </ci> <ci> ν </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> α </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> α </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", "\[Nu]"]], "2"], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]]]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "2"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|