Bessel function of the second kind: Transformations (formula 03.03.16.0007)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

http://functions.wolfram.com/03.03.16.0007.01

Input Form

BesselY[\[Nu], c (d z^n)^m] == ((c d^m z^(m n))^\[Nu]/(c (d z^n)^m)^\[Nu]) BesselY[\[Nu], c d^m z^(m n)] + Cot[Pi \[Nu]] ((c (d z^n)^m)^\[Nu]/(c d^m z^(m n))^\[Nu] - (c d^m z^(m n))^\[Nu]/ (c (d z^n)^m)^\[Nu]) BesselJ[\[Nu], c d^m z^(m n)] /; Element[2 m, Integers] && !Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]]]], "]"]]]]]]]], " ", "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "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> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mfrac> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <cot /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <integers /> </apply> <apply> <notin /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["c_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d_", " ", SuperscriptBox["z_", "n_"]]], ")"]], "m_"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"]], "+", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]], ")"]], "\[Nu]"], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", SuperscriptBox["z", "n"]]], ")"]], "m"]]], ")"]], "\[Nu]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", SuperscriptBox["d", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "n"]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

2001-10-29