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http://functions.wolfram.com/03.21.07.0005.01
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SphericalBesselJ[\[Nu], z] == (I^(-(1/2) - \[Nu])/(Sqrt[2 Pi] Sqrt[z]))
Integrate[E^(I z Cos[t]) Cos[(1/2 + \[Nu]) t], {t, 0, Pi}] /;
Element[1/2 + \[Nu], Integers] && 1/2 + \[Nu] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z", " ", RowBox[List["Cos", "[", "t", "]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]], " ", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> π </mi> </msubsup> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <pi /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <cos /> <ci> t </ci> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z", " ", RowBox[List["Cos", "[", "t", "]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ")"]], " ", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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