html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 EllipticPi

 http://functions.wolfram.com/08.06.06.0094.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/(2 m Sin[z])) (-Log[-4 m Sin[z]^2] + (2 Log[Cos[z/2]^2] - n Sin[z]^2 Sum[((Pochhammer[1/2, j] Sin[z]^(2 j))/((1 + j) j!)) Hypergeometric2F1[1, -j, 1/2 - j, n], {j, 0, Infinity}]) + (1/(4 m)) (2 (1 + n + Cot[z] Csc[z] + Log[Cos[z/2]^2]) - (1 + 2 n) Log[-4 m Sin[z]^2] - n Sin[z]^2 Sum[((Pochhammer[3/2, j] Sin[z]^(2 j))/((1 + j) (1 + j)!)) Hypergeometric2F1[1, -1 - j, -(1/2) - j, n], {j, 0, Infinity}]) + (1/(64 m^2)) (21 + 2 n (7 + 12 n) + 24 n Csc[z]^2 + 6 Cot[z] Csc[z] (3 + 2 Csc[z]^2) + 18 Log[Cos[z/2]^2] - 3 (3 + 4 n + 8 n^2) Log[-4 m Sin[z]^2] - 18 n Sin[z]^2 Sum[(1/((1 + j) (2 + j)!)) (Hypergeometric2F1[1, -2 - j, -(3/2) - j, n] Pochhammer[5/2, j] Sin[z]^(2 j)), {j, 0, Infinity}]) + \[Ellipsis]) + 2 Round[Re[z]/Pi] ((Log[-m]/(2 Sqrt[-m])) (1 + (1 + 2 n)/(4 m) + (3 (3 + 4 n + 8 n^2))/ (64 m^2) + \[Ellipsis]) + ((Sqrt[n] ArcSin[Sqrt[n]])/ (Sqrt[1 - n] Sqrt[-m])) (1 + n/(2 m) + (3 n^2)/(8 m^2) + \[Ellipsis]) + (1/(2 Sqrt[-m])) (4 Log[2] + (1/(2 m)) (-1 + 2 Log[2] + n (-1 + 4 Log[2])) - (21 + 26 n + 28 n^2 - 12 (3 + 4 n + 8 n^2) Log[2])/(64 m^2) + \[Ellipsis])) /; (Abs[m] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], RowBox[List["2", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", 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SqrtBox[RowBox[List["-", "m"]]], " "]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], RowBox[List["4", " ", "m"]]], "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "n"]], "+", RowBox[List["8", " ", SuperscriptBox["n", "2"]]]]], ")"]]]], RowBox[List["64", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", 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["(", RowBox[List["1", "+", FractionBox["n", RowBox[List["2", " ", "m"]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["n", "2"]]], RowBox[List["8", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox[RowBox[List["-", "m"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", 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 MathML Form

 Π ( n ; z m ) 2 Re ( z ) π ( log ( - m ) 2 - m ( 1 + 2 n + 1 4 m + 3 ( 8 n 2 + 4 n + 3 ) 64 m 2 + ) + n sin - 1 ( n ) 1 - n - m ( 1 + n 2 m + 3 n 2 8 m 2 + ) + 1 2 - m ( 4 log ( 2 ) + 2 log ( 2 ) + n ( 4 log ( 2 ) - 1 ) - 1 2 m - 28 n 2 + 26 n - 12 ( 8 n 2 + 4 n + 3 ) log ( 2 ) + 21 64 m 2 + ) ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 2 m sin ( z ) ( - log ( - 4 m sin 2 ( z ) ) + ( 2 log ( cos 2 ( z 2 ) ) - n sin 2 ( z ) j = 0 ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) j ! 2 F 1 ( 1 , - j ; 1 2 - j ; 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["-", "j"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "j"]], 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] ) + 1 4 m ( - n sin 2 ( z ) j = 0 ( 3 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) ( j + 1 ) ! 2 F 1 ( 1 , - j - 1 ; - j - 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", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["1", "2"]]], 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] + 2 ( n + cot ( z ) csc ( z ) + log ( cos 2 ( z 2 ) ) + 1 ) - ( 2 n + 1 ) log ( - 4 m sin 2 ( z ) ) ) + 1 64 m 2 ( 21 + 24 n csc 2 ( z ) + 6 cot ( z ) ( 2 csc 2 ( z ) + 3 ) csc ( z ) + 2 n ( 12 n + 7 ) + 18 log ( cos 2 ( z 2 ) ) - 3 ( 8 n 2 + 4 n + 3 ) log ( - 4 m sin 2 ( z ) ) - 18 n sin 2 ( z ) j = 0 ( 5 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) ( j + 2 ) ! 2 F 1 ( 1 , - j - 2 ; - j - 3 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", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["3", "2"]]], 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] ) ) /; ( "\[LeftBracketingBar]" m "\[RightBracketingBar]" "\[Rule]" ) Π ( n ; z m ) 2 Re ( z ) π ( log ( - m ) 2 - m ( 1 + 2 n + 1 4 m + 3 ( 8 n 2 + 4 n + 3 ) 64 m 2 + ) + n sin - 1 ( n ) 1 - n - m ( 1 + n 2 m + 3 n 2 8 m 2 + ) + 1 2 - m ( 4 log ( 2 ) + 2 log ( 2 ) + n ( 4 log ( 2 ) - 1 ) - 1 2 m - 28 n 2 + 26 n - 12 ( 8 n 2 + 4 n + 3 ) log ( 2 ) + 21 64 m 2 + ) ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 2 m sin ( z ) ( - log ( - 4 m sin 2 ( z ) ) + ( 2 log ( cos 2 ( z 2 ) ) - n sin 2 ( z ) j = 0 ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) j ! 2 F 1 ( 1 , - j ; 1 2 - j ; 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["-", "j"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "j"]], 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] ) + 1 4 m ( - n sin 2 ( z ) j = 0 ( 3 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) ( j + 1 ) ! 2 F 1 ( 1 , - j - 1 ; - j - 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", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["1", "2"]]], 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] + 2 ( n + cot ( z ) csc ( z ) + log ( cos 2 ( z 2 ) ) + 1 ) - ( 2 n + 1 ) log ( - 4 m sin 2 ( z ) ) ) + 1 64 m 2 ( 21 + 24 n csc 2 ( z ) + 6 cot ( z ) ( 2 csc 2 ( z ) + 3 ) csc ( z ) + 2 n ( 12 n + 7 ) + 18 log ( cos 2 ( z 2 ) ) - 3 ( 8 n 2 + 4 n + 3 ) log ( - 4 m sin 2 ( z ) ) - 18 n sin 2 ( z ) j = 0 ( 5 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "j"], Pochhammer] sin 2 j ( z ) ( j + 1 ) ( j + 2 ) ! 2 F 1 ( 1 , - j - 2 ; - j - 3 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", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", FractionBox["3", "2"]]], 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] ) ) /; ( "\[LeftBracketingBar]" m "\[RightBracketingBar]" "\[Rule]" ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02