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InverseJacobiDS






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiDS[z,m] > Differentiation > Fractional integro-differentiation > With respect to m





http://functions.wolfram.com/09.42.20.0010.01









  


  










Input Form





D[InverseJacobiDS[z, m], {m, \[Alpha]}] == Sum[(Pochhammer[1/2, k] Pochhammer[1/2, j] (k + j)! (-1)^(k + j) z^(-2 k - 2 j - 1) m^(k + j - \[Alpha])) (Hypergeometric2F1[(2 k + 2 j + 1)/2, 1/2 + j, (2 k + 2 j + 3)/2, 1/z^2]/ ((2 k + 2 j + 1) Gamma[k + j - \[Alpha] + 1] k! j!)), {k, 0, Infinity}, {j, 0, Infinity}] /; z > 1 && -1 < m < 1










Standard Form





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MathML Form







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</mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, &quot;+&quot;, &quot;j&quot;, &quot;+&quot;, &quot;k&quot;]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;j&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;+&quot;, &quot;j&quot;, &quot;+&quot;, &quot;k&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[&quot;1&quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> m </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> InverseJacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <apply> <plus /> <ci> j </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> j </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> j </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> j </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["InverseJacobiDS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", RowBox[List["2", " ", "j"]], "-", "1"]]], " ", SuperscriptBox["m", RowBox[List["k", "+", "j", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", "1"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], "+", "j"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", "3"]], ")"]]]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "j", "-", "\[Alpha]", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["j", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "m", "<", "1"]]]]]]]]]]










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





2001-10-29





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