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 EllipticPi

 http://functions.wolfram.com/08.06.06.0091.01

 Input Form

 EllipticPi[n, z, m] == 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}] /; Abs[Re[z]] <= Pi/2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", " ", 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", "=", "0"]], "k"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "p"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["p", ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "p"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], "!"]], " "]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["p", "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]]]], ")"]], " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], "+", 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", "!"]], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "q"]]]]], RowBox[List[RowBox[List["(", RowBox[List["j", "-", "q", "-", "1"]], ")"]], "!"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "j", "+", RowBox[List["2", " ", "q"]]]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "j", "-", "q"]], ")"]]]]]], "]"]]]]]]]]]]]]]], " ", ")"]]]], ")"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "k"]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]

 MathML Form

 Π ( n ; z m ) 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 /; "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" π 2 Condition EllipticPi n z m k 0 Pochhammer 1 2 k k -1 n 1 2 n 1 2 z n -1 -1 k -1 1 2 k -1 p 0 k -1 k -1 p Pochhammer p 2 k -1 p 2 2 k -1 p k -1 p -1 -1 p 1 -1 n 1 2 -1 p -1 n 1 2 1 -1 p -1 z p z z j 0 p Binomial p j -1 p -1 j p -1 j 1 -1 n 1 2 j -1 p -1 n 1 2 1 j -1 p -1 q 0 j -1 2 -1 j 2 q 1 q z 2 q Pochhammer -1 j 2 q 2 2 j -1 q -1 j -1 q -1 -1 m -1 k z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[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["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "p"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["p", ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["p", "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["n"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "p"]]]]], ")"]], " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], "+", 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"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "j", "+", RowBox[List["2", " ", "q"]]]]], " ", RowBox[List["q", "!"]], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "q"]]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "j", "+", RowBox[List["2", " ", "q"]]]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "j", "-", "q"]], ")"]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["j", "-", "q", "-", "1"]], ")"]], "!"]]]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "p"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], "!"]]]]]]], RowBox[List["2", " ", RowBox[List["k", "!"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

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

 2007-05-02