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 EllipticPi

 http://functions.wolfram.com/08.06.06.0086.01

 Input Form

 EllipticPi[n, z, m] == ArcTanh[Sqrt[-1 + n] Tan[z]]/ (Sqrt[-1 + n] Sqrt[(-m + n)/n]) + (m/(4 n)) Sum[((Pochhammer[3/2, k] m^k)/(2^k (k + 1)!)) Sum[(-1)^j Binomial[k + 1, j + 1] ((n - 2)/n)^j Sum[(-1)^i Binomial[j + 1, i] (Sum[Binomial[i - 1, 2 q] (Pochhammer[1/2, q]/(2 q!)) (n/(2 - n))^(2 q) (4 z + Tan[2 z] Sum[((p - 1)! Cos[2 z]^(2 p))/Pochhammer[1/2, p], {p, 1, q}]), {q, 0, Floor[(i - 1)/2]}] + Sum[q! Binomial[i - 1, 2 q + 1] (n/(2 - n))^(2 q + 1) Sin[2 z]^(2 q + 1) Sum[Cot[2 z]^(2 p)/(Pochhammer[3/2, q - p] p!), {p, 0, q}], {q, 0, Floor[i/2] - 1}]), {i, 0, j + 1}], {j, 0, k}], {k, 0, Infinity}] + 2 Round[Re[z]/Pi] EllipticPi[n, m] /; Abs[m] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "n"]]]]], "+", RowBox[List[FractionBox["m", RowBox[List["4", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["-", "k"]]], SuperscriptBox["m", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["j", "+", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["n", "-", "2"]], "n"], ")"]], "j"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "+", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "+", "1"]], ",", "i"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["i", "-", "1"]], ",", RowBox[List["2", "q"]]]], "]"]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "q"]], "]"]], RowBox[List["2", RowBox[List["q", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List["2", "q"]]], RowBox[List["(", " ", RowBox[List[RowBox[List["4", "z"]], "+", RowBox[List[RowBox[List["Tan", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], "q"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", "p"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "p"]], "]"]]]]]]]]], ")"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["i", "2"], "]"]], "-", "1"]]], RowBox[List[RowBox[List["q", "!"]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["i", "-", "1"]], ",", RowBox[List[RowBox[List["2", "q"]], "+", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List[RowBox[List["2", "q"]], "+", "1"]]], SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List[RowBox[List["2", "q"]], "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "q"], FractionBox[SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", " ", "p"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["q", "-", "p"]]]], "]"]], RowBox[List["p", "!"]]]]]]]]]]]]], ")"]]]]]]]]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "1"]]]]]]

 MathML Form

 Π ( n ; z m ) tanh - 1 ( n - 1 tan ( z ) ) n - 1 n - m n + m 4 n k = 0 ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 - k m k ( k + 1 ) ! j = 0 k ( - 1 ) j ( k + 1 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["j", "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - 2 n ) j i = 0 j + 1 ( - 1 ) i ( j + 1 i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( q = 0 i 2 - 1 q ! ( i - 1 2 q + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["i", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n 2 - n ) 2 q + 1 sin 2 q + 1 ( 2 z ) p = 0 q cot 2 p ( 2 z ) ( 3 2 ) q - p TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["q", "-", "p"]]], Pochhammer] p ! + q = 0 i - 1 2 ( i - 1 2 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["i", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "q"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 2 ) q TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "q"], Pochhammer] ( n 2 - n ) 2 q 2 q ! ( 4 z + tan ( 2 z ) p = 1 q ( p - 1 ) ! cos 2 p ( 2 z ) ( 1 2 ) p TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "p"], Pochhammer] ) ) + 2 Re ( z ) π Π ( n m ) /; "\[LeftBracketingBar]" m "\[RightBracketingBar]" < 1 Π ( n ; z m ) tanh - 1 ( n - 1 tan ( z ) ) n - 1 n - m n + m 4 n k = 0 ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 - k m k ( k + 1 ) ! j = 0 k ( - 1 ) j ( k + 1 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["j", "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - 2 n ) j i = 0 j + 1 ( - 1 ) i ( j + 1 i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( q = 0 i 2 - 1 q ! ( i - 1 2 q + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["i", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n 2 - n ) 2 q + 1 sin 2 q + 1 ( 2 z ) p = 0 q cot 2 p ( 2 z ) ( 3 2 ) q - p TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["q", "-", "p"]]], Pochhammer] p ! + q = 0 i - 1 2 ( i - 1 2 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["i", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "q"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 2 ) q TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "q"], Pochhammer] ( n 2 - n ) 2 q 2 q ! ( 4 z + tan ( 2 z ) p = 1 q ( p - 1 ) ! cos 2 p ( 2 z ) ( 1 2 ) p TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "p"], Pochhammer] ) ) + 2 Re ( z ) π Π ( n m ) /; "\[LeftBracketingBar]" m "\[RightBracketingBar]" < 1 [/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["ArcTanh", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "n"]]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["m", "k"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["j", "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["n", "-", "2"]], "n"], ")"]], "j"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "+", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "+", "1"]], ",", "i"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["i", "-", "1"]], ",", RowBox[List["2", " ", "q"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "q"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List["2", " ", "q"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "z"]], "+", RowBox[List[RowBox[List["Tan", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], "q"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", " ", "p"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "p"]], "]"]]]]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["q", "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["i", "2"], "]"]], "-", "1"]]], RowBox[List[RowBox[List["q", "!"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["i", "-", "1"]], ",", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "q"], FractionBox[SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", " ", "p"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["q", "-", "p"]]]], "]"]], " ", RowBox[List["p", "!"]]]]]]]]]]]]], ")"]]]]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]], RowBox[List["4", " ", "n"]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "1"]]]]]]]]

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

 2007-05-02