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BesselI




Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[nu,z] > Differentiation > Low-order differentiation > With respect to nu





http://functions.wolfram.com/03.02.20.0020.01









  


  








Input Form





Derivative[1, 0][BesselI][-n - 1/2, z] == (((-1)^n 2 (2 z)^(-(1/2) - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k] (2 n - 2 k)! ((PolyGamma[k + 1/2] - PolyGamma[k - n + 1/2]) Cosh[z] + Cosh[z] CoshIntegral[2 z] - Sinh[z] SinhIntegral[2 z]) z^(2 k), {k, 0, Floor[n/2]}] + (((-1)^n 2 (2 z)^(1/2 - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k + 1] (2 n - 2 k - 1)! ((-(PolyGamma[k + 3/2] - PolyGamma[k - n + 1/2])) Sinh[z] + Cosh[z] SinhIntegral[2 z] - Sinh[z] CoshIntegral[2 z]) z^(2 k), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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<mo> &#8290; </mo> <mrow> <mi> Shi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <apply> <ci> Subscript </ci> <apply> <ci> BesselJ </ci> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 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<apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <ci> CoshIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <ci> SinhIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselI", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]]]], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










Contributed by





Brychkov Yu.A. (2005)


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





2007-05-02