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 EllipticPi

 http://functions.wolfram.com/08.06.06.0001.02

 Input Form

 EllipticPi[n, z, m] \[Proportional] EllipticF[z, m] + (-1)^Round[Re[z]/Pi] (n/3) (AppellF1[3/2, 1/2, 1/2, 5/2, Sin[z]^2, m Sin[z]^2] Sin[z]^3 + (3/5) n AppellF1[5/2, 1/2, 1/2, 7/2, Sin[z]^2, m Sin[z]^2] Sin[z]^5 + (3/7) n^2 AppellF1[7/2, 1/2, 1/2, 9/2, Sin[z]^2, m Sin[z]^2] Sin[z]^7 + \[Ellipsis]) + ((Pi n)/2) (Hypergeometric2F1[3/2, 1/2, 2, m] + (3/4) Hypergeometric2F1[5/2, 1/2, 3, m] n + (5/8) Hypergeometric2F1[7/2, 1/2, 4, m] n^2 + \[Ellipsis]) Round[Re[z]/Pi] /; (n -> 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], FractionBox["n", "3"], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"]]], "+", " ", RowBox[List[FractionBox["3", "5"], " ", "n", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"]]], "+", RowBox[List[FractionBox["3", "7"], " ", SuperscriptBox["n", "2"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["9", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "7"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "n"]], "2"], " ", RowBox[List["(", " ", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", "2", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["3", "4"], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", "3", ",", "m"]], "]"]], "n"]], "+", RowBox[List[FractionBox["5", "8"], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", "4", ",", "m"]], "]"]], SuperscriptBox["n", "2"]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "0"]], ")"]]]]]]

 MathML Form

 Π ( n ; z m ) F ( z m ) + n 3 ( - 1 ) Re ( z ) π ( sin 3 ( z ) F 1 AppellF1 ( 3 2 ; 1 2 , 1 2 ; 5 2 ; sin 2 ( z ) , m sin 2 ( z ) ) + 3 sin 5 ( z ) 5 F 1 AppellF1 ( 5 2 ; 1 2 , 1 2 ; 7 2 ; sin 2 ( z ) , m sin 2 ( z ) ) n + 3 sin 7 ( z ) 7 F 1 AppellF1 ( 7 2 ; 1 2 , 1 2 ; 9 2 ; sin 2 ( z ) , m sin 2 ( z ) ) n 2 + ) + π n 2 Re ( z ) π ( 2 F 1 ( 3 2 , 1 2 ; 2 ; m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["2", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["m", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 3 4 2 F 1 ( 5 2 , 1 2 ; 3 ; m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["3", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["m", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] n + 5 8 2 F 1 ( 7 2 , 1 2 ; 4 ; m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["4", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["m", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] n 2 + ) /; ( n "\[Rule]" 0 ) FormBox RowBox RowBox RowBox Π ( RowBox n ; RowBox z m ) RowBox RowBox F RowBox ( RowBox z m ) + RowBox FractionBox n 3 SuperscriptBox RowBox ( RowBox - 1 ) ErrorBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox RowBox RowBox SuperscriptBox sin 3 ( z ) RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 3 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; FractionBox 5 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) + RowBox FractionBox RowBox 3 RowBox SuperscriptBox sin 5 ( z ) 5 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 5 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; FractionBox 7 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) n + RowBox FractionBox RowBox 3 RowBox SuperscriptBox sin 7 ( z ) 7 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 7 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; FractionBox 9 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) SuperscriptBox n 2 + ) + RowBox FractionBox RowBox π n 2 RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox TagBox TagBox RowBox RowBox SubscriptBox 2 SubscriptBox F 1 RowBox ( RowBox TagBox TagBox RowBox TagBox FractionBox 3 2 Hypergeometric2F1 Rule Editable , TagBox FractionBox 1 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox m Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 + RowBox FractionBox 3 4 TagBox TagBox RowBox RowBox SubscriptBox 2 SubscriptBox F 1 RowBox ( RowBox TagBox TagBox RowBox TagBox FractionBox 5 2 Hypergeometric2F1 Rule Editable , TagBox FractionBox 1 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox 3 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox m Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 n + RowBox FractionBox 5 8 TagBox TagBox RowBox RowBox SubscriptBox 2 SubscriptBox F 1 RowBox ( RowBox TagBox TagBox RowBox TagBox FractionBox 7 2 Hypergeometric2F1 Rule Editable , TagBox FractionBox 1 2 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox 4 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox m Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 SuperscriptBox n 2 + ) /; RowBox ( RowBox n 0 ) 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["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"]]], "+", RowBox[List[FractionBox["3", "5"], " ", "n", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"]]], "+", RowBox[List[FractionBox["3", "7"], " ", SuperscriptBox["n", "2"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["9", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "7"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "n"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", "2", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["3", "4"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", "3", ",", "m"]], "]"]], " ", "n"]], "+", RowBox[List[FractionBox["5", "8"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", "4", ",", "m"]], "]"]], " ", SuperscriptBox["n", "2"]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "0"]], ")"]]]]]]]]

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

 2001-10-29