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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/03.02.20.0002.01









  


  








Input Form





Derivative[1, 0][BesselI][\[Nu], z] == (-(1/((\[Nu] + 1) Gamma[\[Nu] + 2]))) (z/2)^(\[Nu] + 2) HypergeometricPFQ[{{}, {1}, {1, \[Nu] + 1}}, {{2, \[Nu] + 2}, {}, {\[Nu] + 2}}, z^2/4, z^2/4] + BesselI[\[Nu], z] (Log[z] - Log[2] - PolyGamma[\[Nu] + 1])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "BesselI", "]"]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "2"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "+", "2"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["\[Nu]", "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["\[Nu]", "+", "2"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "+", "2"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation encoding='Mathematica'> TagBox[&quot;I&quot;, BesselI] </annotation> </semantics> <mi> &#957; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <mi> I </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mn> 0 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 0 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ; </mo> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> &#8290; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> I </ms> <ci> BesselI </ci> </apply> <ms> &#957; </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ci> Derivative </ci> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> I </ms> <ms> &#957; </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; 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</ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> <ms> ] </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "+", "2"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["\[Nu]", "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["\[Nu]", "+", "2"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", "+", "2"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "2"]], "]"]]]]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]]]]]]]]]]


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





2001-10-29