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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Differentiation > Fractional integro-differentiation > With respect to nu





http://functions.wolfram.com/07.06.20.0007.01









  


  










Input Form





D[Fibonacci[\[Nu], z], {\[Nu], \[Alpha]}] == (1/(2 Sqrt[4 + z^2])) ((2 w^\[Nu] (1 - GammaRegularized[-\[Alpha], \[Nu] Log[w]]) (\[Nu] Log[w])^\[Alpha])/\[Nu]^\[Alpha] - ((1 - GammaRegularized[-\[Alpha], \[Nu] ((-I) Pi - Log[w])]) (\[Nu] ((-I) Pi - Log[w]))^\[Alpha])/(E^(I Pi \[Nu]) w^\[Nu] \[Nu]^\[Alpha]) - (E^(I Pi \[Nu]) (1 - GammaRegularized[-\[Alpha], \[Nu] (I Pi - Log[w])]) (\[Nu] (I Pi - Log[w]))^\[Alpha])/ (w^\[Nu] \[Nu]^\[Alpha])) /; w == (z + Sqrt[z^2 + 4])/2










Standard Form





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MathML Form







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</mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> w </ci> <ci> &#957; </ci> </apply> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> GammaRegularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> &#957; </ci> <apply> <ln /> <ci> w </ci> </apply> </apply> <ci> &#945; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> GammaRegularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> GammaRegularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> </apply> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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