|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/03.21.23.0008.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Sum[Sin[(1 + 2 k) t] SphericalBesselJ[2 k + 1/2, x], {k, 0, Infinity}] ==
Sqrt[Pi/2] (1/(2 Sqrt[x])) Sin[x Sin[t]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", "t"]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", FractionBox["1", "2"]]], ",", "x"]], "]"]]]]]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox[RowBox[List["1", " "]], RowBox[List["2", " ", SqrtBox["x"]]]], RowBox[List["Sin", "[", RowBox[List["x", " ", RowBox[List["Sin", "[", "t", "]"]]]], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> j </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> x </mi> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> t </ci> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> x </ci> <apply> <sin /> <ci> t </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k_"]]]], ")"]], " ", "t_"]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k_"]], "+", FractionBox["1", "2"]]], ",", "x_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["Sin", "[", RowBox[List["x", " ", RowBox[List["Sin", "[", "t", "]"]]]], "]"]]]], RowBox[List["2", " ", SqrtBox["x"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|