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 EllipticPi

 http://functions.wolfram.com/08.06.20.0008.02

 Input Form

 D[EllipticPi[n, z, m], {z, p}] == KroneckerDelta[p] EllipticPi[n, z, m] + (((I^(p + 1) 2^(p - 2) E^(2 I z) Sqrt[Pi])/(Sqrt[1 - n] Sqrt[1 - m Sin[z]^2])) Sqrt[(2 (1 + Sqrt[1 - m]) + (E^(2 I z) - 1) m)/ Sqrt[1 - m]] Sum[E^(2 I k z) StirlingS2[p - 1, k] Sum[((Binomial[k, j] Pochhammer[j - k, k - j])/Gamma[1/2 - j]) (m/(2 - 2 Sqrt[1 - m] + (E^(2 I z) - 1) m))^j ((E^(2 I z) - 1 + (2 (1 + Sqrt[1 - n]))/n)^(j - k - 1) - (E^(2 I z) - 1 + (2 (1 - Sqrt[1 - n]))/n)^(j - k - 1)) AppellF1[1/2, 1/2, -(3/2), 1/2 - j, (2 Sqrt[1 - m] - 2 + m (1 - E^(2 I z)))/(4 Sqrt[1 - m]), (2 Sqrt[1 - m] - 2 + m (1 - E^(2 I z)))/(2 Sqrt[1 - m] - 2 + m)], {j, 0, k}], {k, 0, p - 1}])/((E^(2 I z) (m - 2 (1 + Sqrt[1 - m])))/m)^(3/2) /; Element[p, Integers] && p >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "p"]], "}"]]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "p", "]"]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["p", "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["p", "-", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], SqrtBox["\[Pi]"], " "]], RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], SqrtBox[RowBox[List["1", "-", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]]]], ")"]]]], "m"], ")"]], RowBox[List[RowBox[List["-", "3"]], "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "z"]]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["p", "-", "1"]], ",", "k"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["k", "-", "j"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]]], SuperscriptBox[RowBox[List["(", FractionBox["m", RowBox[List["2", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]]], ")"]], "j"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "n"]]]]], ")"]]]], "n"]]], ")"]], RowBox[List["j", "-", "k", "-", "1"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "n"]]]]], ")"]]]], "n"]]], ")"]], RowBox[List["j", "-", "k", "-", "1"]]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "2", "+", RowBox[List["m", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], " ", ")"]]]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "2", "+", RowBox[List["m", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], " ", ")"]]]]]], RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "2", "+", "m"]]]]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 p Π ( n ; z m ) z p Π ( n ; z m ) δ KroneckerDelta p + p + 1 2 p - 2 2 z π 1 - n 1 - m sin 2 ( z ) 2 ( 1 - m + 1 ) + ( 2 z - 1 ) m 1 - m ( 2 z ( m - 2 ( 1 - m + 1 ) ) m ) - 3 / 2 k = 0 p - 1 2 k z 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] p - 1 ( k ) j = 0 k ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( j - k ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k"]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] Γ ( 1 2 - j ) ( m ( 2 z - 1 ) m - 2 1 - m + 2 ) j ( ( 2 ( 1 + 1 - n ) n + 2 z - 1 ) j - k - 1 - ( 2 ( 1 - 1 - n ) n + 2 z - 1 ) j - k - 1 ) F 1 AppellF1 ( 1 2 ; 1 2 , - 3 2 ; 1 2 - j ; m ( 1 - 2 z ) + 2 1 - m - 2 4 1 - m , m ( 1 - 2 z ) + 2 1 - m - 2 m + 2 1 - m - 2 ) /; p Condition z p EllipticPi n z m EllipticPi n z m KroneckerDelta p p 1 2 p -2 2 z 1 2 1 -1 n 1 2 1 -1 m z 2 1 2 -1 2 1 -1 m 1 2 1 2 z -1 m 1 -1 m 1 2 -1 1 2 2 z m -1 2 1 -1 m 1 2 1 m -1 -3 2 k 0 p -1 2 k z StirlingS2 p -1 k j 0 k Binomial k j Pochhammer j -1 k k -1 j Gamma 1 2 -1 j -1 m 2 z -1 m -1 2 1 -1 m 1 2 2 -1 j 2 1 1 -1 n 1 2 n -1 2 z -1 j -1 k -1 -1 2 1 -1 1 -1 n 1 2 n -1 2 z -1 j -1 k -1 AppellF1 1 2 1 2 -1 3 2 1 2 -1 j m 1 -1 2 z 2 1 -1 m 1 2 -2 4 1 -1 m 1 2 -1 m 1 -1 2 z 2 1 -1 m 1 2 -2 m 2 1 -1 m 1 2 -2 -1 p [/itex]

 Rule Form

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

 2001-10-29