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BesselY




Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/03.03.16.0014.01









  


  








Input Form





BesselY[\[Nu], Sqrt[z^2]] == (Sqrt[z^2]/z)^\[Nu] (BesselY[\[Nu], z] + (2/Pi) (Log[Sqrt[z^2]] - Log[z]) BesselJ[\[Nu], z]) /; Element[\[Nu], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", SqrtBox[SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox["2", "\[Pi]"], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]








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> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <semantics> <mrow> <mi> &#8484; </mi> <mo> . </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[DoubleStruckCapitalZ]&quot;, &quot;.&quot;]], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <semantics> <mrow> <mi> &#8484; </mi> <mo> . </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[DoubleStruckCapitalZ]&quot;, &quot;.&quot;]], Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", SqrtBox[SuperscriptBox["z_", "2"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Pi]"]]], ")"]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]


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





2007-05-02