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 EllipticPi

 http://functions.wolfram.com/08.03.06.0048.01

 Input Form

 EllipticPi[n, m] == (Log[-m]/(2 Sqrt[-m])) Sum[((Pochhammer[1/2, k]^2/k!^2) Hypergeometric2F1[1, -k, 1/2 - k, n])/ m^k, {k, 0, Infinity}] + ((Sqrt[n] ArcSin[Sqrt[n]])/ (Sqrt[1 - n] Sqrt[-m])) Sum[(n^k Pochhammer[1/2, k])/k!/m^k, {k, 0, Infinity}] + (1/(2 Sqrt[-m])) (Log[16] + Sum[((Pochhammer[1/2, k]^2/k!) (((-n)^k/Pochhammer[1/2 - k, k]) (Log[16] - Sum[2/(i + k), {i, 0, k - 1}] + 1/k) + Sum[((-n)^j/(Pochhammer[1/2 - k, j] (k - j)!)) (1/k + 1/(k - j) - Sum[2/(i + k), {i, 0, k - 1}] - Sum[2/(i + k - j), {i, 0, k - j - 1}] + Log[16]), {j, 0, k - 1}]))/m^k, {k, 1, Infinity}])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], RowBox[List["2", SqrtBox[RowBox[List["-", "m"]]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "n"]], "]"]], SuperscriptBox["m", RowBox[List["-", "k"]]]]]]]]], " ", "+", RowBox[List[FractionBox[RowBox[List[SqrtBox["n"], RowBox[List["ArcSin", "[", SqrtBox["n"], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], SqrtBox[RowBox[List["-", "m"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["n", "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], RowBox[List[" ", RowBox[List["k", "!"]]]]], SuperscriptBox["m", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox[RowBox[List["-", "m"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "16", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], RowBox[List[RowBox[List["k", "!"]], " "]]], SuperscriptBox["m", RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "k"]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "16", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k"]]]]], "+", FractionBox["1", "k"]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "j"], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "j"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " "]]], RowBox[List["(", RowBox[List[FractionBox["1", "k"], "+", FractionBox["1", RowBox[List["k", "-", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "j", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k", "-", "j"]]]]], "+", RowBox[List["Log", "[", "16", "]"]]]], ")"]]]]]]]], ")"]]]]]]]], ")"]]]]]]]]]]

 MathML Form

 Π ( n m ) n sin - 1 ( n ) 1 - n - m k = 0 n k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! m - k + log ( - m ) 2 - m k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ( k ! ) 2 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["n", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] m - k + 1 2 - m ( log ( 16 ) + k = 1 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 m - k k ! ( ( - n ) k ( 1 2 - k ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "k"]], ")"]], "k"], Pochhammer] ( log ( 16 ) - i = 0 k - 1 2 i + k + 1 k ) + j = 0 k - 1 ( - n ) j ( 1 2 - k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "k"]], ")"]], "j"], Pochhammer] ( k - j ) ! ( log ( 16 ) - i = 0 - j + k - 1 2 i - j + k - i = 0 k - 1 2 i + k + 1 k - j + 1 k ) ) ) EllipticPi n m n 1 2 n 1 2 1 -1 n 1 2 -1 m 1 2 -1 k 0 n k Pochhammer 1 2 k k -1 m -1 k -1 m 2 -1 m 1 2 -1 k 0 Pochhammer 1 2 k 2 k 2 -1 Hypergeometric2F1 1 -1 k 1 2 -1 k n m -1 k 1 2 -1 m 1 2 -1 16 k 1 Pochhammer 1 2 k 2 m -1 k k -1 -1 n k Pochhammer 1 2 -1 k k -1 16 -1 i 0 k -1 2 i k -1 1 k -1 j 0 k -1 -1 n j Pochhammer 1 2 -1 k j k -1 j -1 16 -1 i 0 -1 j k -1 2 i -1 j k -1 -1 i 0 k -1 2 i k -1 1 k -1 j -1 1 k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "n"]], "]"]], " ", SuperscriptBox["m", RowBox[List["-", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["n"], " ", RowBox[List["ArcSin", "[", SqrtBox["n"], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["n", "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["m", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["Log", "[", "16", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", SuperscriptBox["m", RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "16", "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k"]]]]], "+", FractionBox["1", "k"]]], ")"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "k"]], "]"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "k"], "+", FractionBox["1", RowBox[List["k", "-", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "j", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k", "-", "j"]]]]], "+", RowBox[List["Log", "[", "16", "]"]]]], ")"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]]]]]]]]

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

 2007-05-02