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http://functions.wolfram.com/03.04.16.0013.01
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BesselK[\[Nu], Sqrt[z^2]] == (z^\[Nu] BesselK[\[Nu], z])/(z^2)^(\[Nu]/2) -
(1/2) Pi (-z^\[Nu]/(z^2)^(\[Nu]/2) + (z^2)^(\[Nu]/2)/z^\[Nu])
BesselI[\[Nu], z] Csc[Pi \[Nu]] /; !Element[\[Nu], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", SqrtBox[SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["z", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]
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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> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mi> ν </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselK </ci> <ci> ν </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> ν </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", SqrtBox[SuperscriptBox["z_", "2"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["z", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "2"]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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