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 EllipticPi

 http://functions.wolfram.com/08.06.06.0090.01

 Input Form

 EllipticPi[n, z, m] == (-1)^Round[Re[z]/Pi] Sum[(Pochhammer[1/2, k]/k!) ((n - 1)^(-k - 1) Sqrt[n] ArcTanh[Sqrt[n] Sin[z]] + (1/(2 k!)) Sum[(((-1)^(k - p) Pochhammer[p, 2 (k - p)])/(2^(2 k - p) (k - p)!)) ((-1)^p p! ((1 - Sqrt[n])^(-1 - p) + (1 + Sqrt[n])^(-1 - p)) ArcTanh[Sin[z]] + Tan[z] Sec[z] Sum[Binomial[p, j] (-1)^(p - j) (p - j)! ((1 - Sqrt[n])^(-1 + j - p) + (1 + Sqrt[n])^ (-1 + j - p)) Sum[((2^(1 - j + 2 q) q! Tan[z]^(2 q))/ (j - q - 1)!) Pochhammer[2 - j + 2 q, 2 (-1 + j - q)], {q, 0, j - 1}], {j, 0, p}]), {p, 0, k}]) (m - 1)^k, {k, 0, Infinity}] - Round[Re[z]/Pi] (Log[1 - m] Sum[(((1 - m)^j Pochhammer[1/2, j]^2)/j!^2) Hypergeometric2F1[1, 1/2 + j, 1/2, n], {j, 0, Infinity}] - Sum[(((1 - m)^k Pochhammer[1/2, k]^2)/k!^2) Sum[(Pochhammer[1/2 + k, s]/Pochhammer[1/2, s]) n^s (-PolyGamma[1/2 + k] + 2 PolyGamma[1 + k] - PolyGamma[1/2 + k + s]), {s, 0, Infinity}], {k, 0, Infinity}]) /; !Element[(Pi + 2 Re[z])/(4 Pi), Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " "]], RowBox[List[" ", RowBox[List["k", "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", "1"]]], SqrtBox["n"], " ", RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], " ", "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["k", "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", 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"+", RowBox[List[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["Sec", "[", "z", "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "p"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["p", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "j"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "j", "-", "p"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "j", "-", "p"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "j", "+", RowBox[List["2", " ", "q"]]]]], " ", RowBox[List["q", "!"]], " ", 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RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]]], " ", SuperscriptBox["n", "s"], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "s"]], "]"]]]], ")"]]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "]"]]]]]]

 MathML Form

 Π ( n ; z m ) ( - 1 ) Re ( z ) π k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! ( n tanh - 1 ( n sin ( z ) ) ( n - 1 ) - k - 1 + 1 2 k ! p = 0 k ( - 1 ) k - p ( p ) 2 ( k - p ) TagBox[SubscriptBox[RowBox[List["(", "p", ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]], Pochhammer] 2 2 k - p ( k - p ) ! ( ( - 1 ) p ( ( 1 - n ) - p - 1 + ( n + 1 ) - p - 1 ) tanh - 1 ( sin ( z ) ) p ! + tan ( z ) sec ( z ) j = 0 p ( p j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["p", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) p - j ( p - j ) ! ( ( 1 - n ) j - p - 1 + ( n + 1 ) j - p - 1 ) q = 0 j - 1 ( 2 - j + 2 q + 1 q ! tan 2 q ( z ) ) ( - j + 2 q + 2 ) 2 ( j - q - 1 ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", RowBox[List["2", " ", "q"]], "+", "2"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "q", "-", "1"]], ")"]]]]], Pochhammer] ( j - q - 1 ) ! ) ) ( m - 1 ) k - Re ( z ) π ( log ( 1 - m ) j = 0 ( 1 - m ) j ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] 2 ( j ! ) 2 2 F 1 ( 1 , j + 1 2 ; 1 2 ; n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["n", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - k = 0 ( 1 - m ) k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ( k ! ) 2 s = 0 ( k + 1 2 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "+", FractionBox["1", "2"]]], ")"]], "s"], Pochhammer] ( 2 ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + s + 1 2 ) ) n s ( 1 2 ) s TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "s"], Pochhammer] ) /; ¬ 2 Re ( z ) + π 4 π TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] FormBox RowBox RowBox RowBox Π ( RowBox n ; RowBox z m ) RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) ErrorBox RowBox FractionBox RowBox Re ( z ) π RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer RowBox k ! RowBox ( RowBox RowBox SqrtBox n RowBox SuperscriptBox tanh RowBox - 1 ( RowBox SqrtBox n RowBox sin ( z ) ) SuperscriptBox RowBox ( RowBox n - 1 ) RowBox RowBox - k - 1 + RowBox FractionBox 1 RowBox 2 RowBox k ! RowBox UnderoverscriptBox RowBox p = 0 k RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox k - p TagBox SubscriptBox RowBox ( p ) RowBox 2 RowBox ( RowBox k - p ) Pochhammer RowBox SuperscriptBox 2 RowBox RowBox 2 k - p RowBox RowBox ( RowBox k - p ) ! RowBox ( RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) p RowBox ( RowBox SuperscriptBox RowBox ( RowBox 1 - SqrtBox n ) RowBox RowBox - p - 1 + SuperscriptBox RowBox ( RowBox SqrtBox n + 1 ) RowBox RowBox - p - 1 ) RowBox SuperscriptBox tanh RowBox - 1 ( RowBox sin ( z ) ) RowBox p ! + RowBox RowBox tan ( z ) RowBox sec ( z ) RowBox UnderoverscriptBox RowBox j = 0 p RowBox TagBox RowBox ( GridBox TagBox p Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable SuperscriptBox RowBox ( RowBox - 1 ) RowBox p - j RowBox RowBox ( RowBox p - j ) ! RowBox ( RowBox SuperscriptBox RowBox ( RowBox 1 - SqrtBox n ) RowBox j - p - 1 + SuperscriptBox RowBox ( RowBox SqrtBox n + 1 ) RowBox j - p - 1 ) RowBox UnderoverscriptBox RowBox q = 0 RowBox j - 1 FractionBox RowBox RowBox ( RowBox SuperscriptBox 2 RowBox RowBox - j + RowBox 2 q + 1 RowBox q ! RowBox SuperscriptBox tan RowBox 2 q ( z ) ) TagBox SubscriptBox RowBox ( RowBox RowBox - j + RowBox 2 q + 2 ) RowBox 2 RowBox ( RowBox j - q - 1 ) Pochhammer RowBox RowBox ( RowBox j - q - 1 ) ! ) ) SuperscriptBox RowBox ( RowBox m - 1 ) k - RowBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox RowBox RowBox log ( RowBox 1 - m ) RowBox UnderoverscriptBox RowBox j = 0 RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox 1 - m ) j SuperscriptBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) j Pochhammer 2 SuperscriptBox RowBox ( RowBox j ! ) 2 TagBox TagBox RowBox RowBox SubscriptBox 2 SubscriptBox F 1 RowBox ( RowBox TagBox TagBox RowBox TagBox 1 HypergeometricPFQ Rule Editable , TagBox RowBox j + FractionBox 1 2 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox TagBox FractionBox 1 2 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox n HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ - RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox 1 - m ) k SuperscriptBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer 2 SuperscriptBox RowBox ( RowBox k ! ) 2 RowBox UnderoverscriptBox RowBox s = 0 FractionBox RowBox TagBox SubscriptBox RowBox ( RowBox k + FractionBox 1 2 ) s Pochhammer RowBox ( RowBox RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox k + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox k + FractionBox 1 2 ) - RowBox TagBox ψ PolyGamma ( RowBox k + s + FractionBox 1 2 ) ) SuperscriptBox n s TagBox SubscriptBox RowBox ( FractionBox 1 2 ) s Pochhammer ) /; RowBox ¬ RowBox FractionBox RowBox RowBox 2 RowBox Re ( z ) + π RowBox 4 π TagBox Function TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", "1"]]], " ", SqrtBox["n"], " ", RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", 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RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "s"]], "]"]]]], ")"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]]]], ")"]]]]]], "/;", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02