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BesselI






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[nu,z] > Series representations > Generalized power series > Expansions at nu==+-n





http://functions.wolfram.com/03.02.06.0022.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] BesselI[n, z] + (Sum[(((-1)^(n - k - 1) (n - k - 1)!)/k!) (z/2)^(2 k - n), {k, 0, n - 1}] - (-1)^n BesselK[n, z] + ((n!/2) Sum[((-1)^k/((n - k) k!)) BesselI[k, z] (z/2)^k, {k, 0, n - 1}])/ (-(z/2))^n + (1/n!) (z/2)^n Sum[(1/j) HypergeometricPFQ[{j}, {1 + j, 1 + n}, z^2/4], {j, 1, n}]) (\[Nu] + n) + \[Ellipsis] /; (\[Nu] -> -n) && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " "]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", "k"]], "-", "n"]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["BesselK", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["n", "!"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["BesselI", "[", RowBox[List["k", ",", "z"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[FractionBox["1", "j"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "j", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["1", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]]]], ")"]], RowBox[List["(", RowBox[List["\[Nu]", "+", "n"]], ")"]]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










MathML Form







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</mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> j </mi> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> ; </mo> <mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[&quot;j&quot;, HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]], RowBox[List["k", "!"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["BesselK", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["BesselI", "[", RowBox[List["k", ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "j", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["1", "+", "n"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]], "j"]]]]], RowBox[List["n", "!"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "n"]], ")"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02