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 EllipticPi

 http://functions.wolfram.com/08.06.20.0018.01

 Input Form

 D[EllipticPi[n, z, m], {m, p}] == (((ArcTanh[Sqrt[-1 + n] Tan[z]] Sqrt[Cos[z]^2] Sec[z])/Sqrt[-1 + n]) Hypergeometric2F1Regularized[1/2, 1, 1 - p, m/n] - ((m Sin[z])/(4 n)) Sum[(((k + 1)! Pochhammer[3/2, k] m^k)/ (2^k (k - p + 1)!)) Sum[(1/(k - j)!) ((n - 2)/n)^j Sum[((-1)^(i + j - 1)/(i! (j - i + 1)!)) (Sum[((Binomial[-1 + i, 2 q] Pochhammer[1/2, q])/q!) (n/(2 - n))^(2 q) ((2 ArcSin[Sin[z]])/Sin[z] + (Sqrt[Cos[z]^2]/ Cos[2 z]) Sum[((p - 1)! Cos[2 z]^(2 p))/Pochhammer[1/2, p], {p, 1, q}]), {q, 0, Floor[(i - 1)/2]}] + ((2 n Sqrt[Cos[z]^2])/(2 - n)) Sum[q! Binomial[-1 + i, 1 + 2 q] Cos[2 z]^(2 q) (n/(2 - n))^(2 q) Sum[Tan[2 z]^(2 p)/ ((q - p)! Pochhammer[3/2, p]), {p, 0, q}], {q, 0, -1 + Floor[i/2]}]), {i, 0, 1 + j}], {j, 0, k}], {k, 0, Infinity}])/ m^p /; Element[p, Integers] && p >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "p"]], "}"]]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["m", RowBox[List["-", "p"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]], " ", RowBox[List["Sec", "[", "z", "]"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " "]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[FractionBox["1", "2"], ",", "1", ",", RowBox[List["1", "-", "p"]], ",", FractionBox["m", "n"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["Sin", "[", "z", "]"]]]], RowBox[List["4", " ", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["-", "k"]]], SuperscriptBox["m", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "p", "+", "1"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["n", "-", "2"]], "n"], ")"]], "j"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["1", "+", "j"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "+", "j", "-", "1"]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "+", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "i"]], ",", RowBox[List["2", " ", "q"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "q"]], "]"]]]], RowBox[List["q", "!"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List["2", " ", "q"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["ArcSin", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], RowBox[List["Sin", "[", "z", "]"]]], " ", "+", RowBox[List[FractionBox[SqrtBox[SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]], RowBox[List["Cos", "[", 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[FractionBox[RowBox[List["2", "n", SqrtBox[SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]]], RowBox[List["2", "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Floor", "[", FractionBox["i", "2"], "]"]]]]], RowBox[List[RowBox[List["q", "!"]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "i"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "q"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["n", RowBox[List["2", "-", "n"]]], ")"]], RowBox[List["2", " ", "q"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], "q"], FractionBox[SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List["2", "p"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "p"]], "]"]]]]]]]]]]]]]]], ")"]]]]]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 p Π ( n ; z m ) m p m - p ( tanh - 1 ( n - 1 tan ( z ) ) cos 2 ( z ) sec ( z ) n - 1 2 F ~ 1 ( 1 2 , 1 ; 1 - p ; m n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "p"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox["m", "n"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - m sin ( z ) 4 n k = 0 ( k + 1 ) ! ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 - k m k ( k - p + 1 ) ! j = 0 k ( n - 2 n ) j ( k - j ) ! i = 0 j + 1 ( - 1 ) i + j - 1 i ! ( j - i + 1 ) ! ( 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] q ! ( n 2 - n ) 2 q ( 2 sin - 1 ( sin ( z ) ) sin ( z ) + cos 2 ( z ) cos ( 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 n cos 2 ( z ) 2 - n 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]] cos 2 q ( 2 z ) ( n 2 - n ) 2 q p = 0 q tan 2 p ( 2 z ) ( q - p ) ! ( 3 2 ) p TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "p"], Pochhammer] ) ) /; p Condition m p EllipticPi n z m m -1 p n -1 1 2 z z 2 1 2 z n -1 1 2 -1 Hypergeometric2F1Regularized 1 2 1 1 -1 p m n -1 -1 m z 4 n -1 k 0 k 1 Pochhammer 3 2 k 2 -1 k m k k -1 p 1 -1 j 0 k n -2 n -1 j k -1 j -1 i 0 j 1 -1 i j -1 i j -1 i 1 -1 q 0 i -1 2 -1 Binomial i -1 2 q Pochhammer 1 2 q q -1 n 2 -1 n -1 2 q 2 z z -1 z 2 1 2 2 z -1 p 1 q p -1 2 z 2 p Pochhammer 1 2 p -1 2 n z 2 1 2 2 -1 n -1 q 0 i 2 -1 -1 q Binomial i -1 2 q 1 2 z 2 q n 2 -1 n -1 2 q p 0 q 2 z 2 p q -1 p Pochhammer 3 2 p -1 p [/itex]

 Rule Form

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

 2007-05-02