BesselI[\[Nu], z] == (z^\[Nu]/(2^\[Nu] (Sqrt[Pi] Gamma[\[Nu] + 1/2]))) Integrate[(1 - t^2)^(v - 1/2)/E^(z t), {t, -1, -1}] /; Re[\[Nu]] > -(1/2)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " "]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "]"]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], RowBox[List["-", "1"]]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], RowBox[List["v", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "t"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]

<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> I </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> v </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselI </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> -1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> v </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"]]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], RowBox[List["-", "1"]]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], RowBox[List["v", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "t"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]]]

2001-10-29