Incomplete elliptic integral of the third kind: Series representations (formula 08.06.06.0030)

EllipticPi

$Mathematica Notation$

Elliptic Integrals

EllipticPi[n,z,m]

Series representations

Generalized power series

Expansions on branch cuts

Formulas on real axis for real m, n

For m<1<n,Pi(u+1/2)<xu+1)-csc^-1(n^1/2)/;u ∈ Z

http://functions.wolfram.com/08.06.06.0030.01

Input Form

EllipticPi[n, z, m] == EllipticPi[n, x, m] - ((Pi I Sqrt[n])/(2 Sqrt[n - 1] Sqrt[n - m])) (1 - Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]] Sum[(1/p!) Sum[(1/j!) Sum[Binomial[j, Subscript[k, 1]] Sum[((-1)^Subscript[k, 1] 2^(Subscript[k, 1] - j) Sin[x]^Subscript[k, 1] (2 Subscript[k, 2] + Subscript[k, 1] - j)^ p Binomial[j - Subscript[k, 1], Subscript[k, 2]] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)! (2 Sin[x])^ (j - 2 i - 1))) Sum[(-1)^Subscript[i, 1] KroneckerDelta[ Subscript[i, 1] + Subscript[i, 2] + Subscript[i, 3] - i] Multinomial[Subscript[i, 1], Subscript[i, 2], Subscript[i, 3]] n^Subscript[i, 1] m^Subscript[i, 3] Pochhammer[ -Subscript[i, 1], Subscript[i, 1]] Pochhammer[1/2, Subscript[i, 2]] Pochhammer[1/2, Subscript[i, 3]] (1 - n Sin[x]^2)^(-1 - Subscript[i, 1]) Cos[x]^ (-2 Subscript[i, 2] - 1) (1 - m Sin[x]^2)^(-(1/2) - Subscript[i, 3]), {Subscript[i, 1], 0, i}, {Subscript[i, 2], 0, i}, {Subscript[i, 3], 0, i}], {i, 0, j - 1}])/E^((1/2) I ((p - 2 Subscript[k, 2] - Subscript[k, 1] + j) Pi + 2 (2 Subscript[k, 2] + Subscript[k, 1] - j) x)), {Subscript[k, 2], 0, j - Subscript[k, 1]}], {Subscript[k, 1], 0, j - 1}], {j, 1, p}] (z - x)^p, {p, 1, Infinity}] /; Element[x, Reals] && Element[m, Reals] && Element[n, Reals] && m < 1 < n && Pi/2 + Pi u < x < Pi (u + 1) - ArcCsc[Sqrt[n]] && Element[u, Integers]

Standard Form

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MathML Form

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<mi> p </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "1"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "-", SubscriptBox["k", "1"]]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mfrac> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "j"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["sin", "(", "x", ")"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", "i"]], "-", "1"]]]]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> i </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mi> i </mi> </mrow> </msub> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["i", "1"], "+", SubscriptBox["i", "2"], "+", SubscriptBox["i", "3"]]], ";", SubscriptBox["i", "1"]]], ",", SubscriptBox["i", "2"], ",", SubscriptBox["i", "3"]]], ")"]], Multinomial, Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> n </mi> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msup> <mo> ⁢ </mo> <msup> <mi> m </mi> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["i", "1"]]], ")"]], SubscriptBox["i", "1"]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 2 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], SubscriptBox["i", "2"]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], SubscriptBox["i", "3"]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msub> <mi> i 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Rule Form

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

2007-05-02

EllipticPi[n,m]