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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Series representations > Generalized power series > Expansions on branch cuts





http://functions.wolfram.com/08.03.06.0036.01









  


  










Input Form





EllipticPi[n, m] == Sum[((Pochhammer[1/2, k + j]/(j! k! (2 k + 2 j + 1))) AppellF1[1/2 + k + j, 1, 1/2, 3/2 + k + j, n/x, 1/x] (m - x)^j)/x^j, {j, 0, Infinity}, {k, 0, Infinity}]/x^2^(-1) + (((I Sqrt[x - 1])/((n - 1) x)) Sum[(((-1)^j Pochhammer[k + 1/2, j] Pochhammer[1/2, k])/(j! k!)) x^(-j - k) AppellF1[1/2, 1 + k, 1, 3/2, 1 - 1/x, (n (x - 1))/((n - 1) x)] (m - x)^j, {j, 0, Infinity}, {k, 0, Infinity}])/E^(I Pi Floor[Arg[x - m]/(2 Pi)]) /; Element[x, Reals] && x > 1










Standard Form





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MathML Form







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<mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> x </mi> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> 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encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> j </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> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <ci> 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Rule Form





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





2007-05-02





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