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variants of this functions
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[nu,z] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/03.20.20.0015.01









  


  










Input Form





D[KelvinKer[\[Nu], z], {z, n}] == 2^(-1 - 3 (n/2)) (I - 1)^n (Sum[Binomial[n, 2 k] (I (1 - I^n) KelvinKei[4 k - n + \[Nu], z] + (1 + I^n) KelvinKer[4 k - n + \[Nu], z]), {k, 0, Floor[n/2]}] + Sum[Binomial[n, 2 k + 1] ((-I) (1 - I^n) KelvinKei[2 + 4 k - n + \[Nu], z] - (1 + I^n) KelvinKer[2 + 4 k - n + \[Nu], z]), {k, 0, Floor[(n - 1)/2]}]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List[&quot;2&quot;, &quot; 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</mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> <apply> <plus /> <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> <ci> Binomial </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <imaginaryi /> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> KelvinKei </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox[RowBox[List["3", " ", "n"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "k"]], "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02