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/08.06.06.0097.01

 Input Form

 EllipticPi[n, z, m] == (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/ (2 m Sin[z])) Sum[((Pochhammer[1/2, k] n^j m^(-j - k))/k!) ((Pochhammer[3/2, j + k]/(2 m Sin[z]^2)) HypergeometricPFQRegularized[ {1, 1, 3/2 + j + k}, {2 + j + k, 2}, Csc[z]^2/m] - (Pochhammer[1/2, j + k]/(j + k)!) (Log[(-m) Sin[z]^2] - PolyGamma[1/2 + j + k] + PolyGamma[1 + j + k]) + 2 m Sin[z]^2 Sum[(Pochhammer[-(1/2), -i + j + k] m^i Sin[z]^(2 i))/ ((1 + i) (-1 - i + j + k)!), {i, 0, -1 + j + k}]), {k, 0, Infinity}, {j, 0, Infinity}] + 2 Round[Re[z]/Pi] EllipticPi[n, m]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", 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[" ", RowBox[List["2", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], SuperscriptBox["n", "j"], SuperscriptBox["m", RowBox[List[RowBox[List["-", "j"]], "-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["j", "+", "k"]]]], "]"]], RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "j", "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "j", "+", "k"]], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]], "]"]]]], " ", "-", " ", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["j", "+", "k"]]]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", "k"]], "]"]]]], ")"]]]], " ", "+", RowBox[List["2", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "j", "+", "k"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "j", "+", "k"]]]], "]"]], " ", SuperscriptBox["m", "i"], SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "i", "+", "j", "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]]]]]], " ", "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]]]]

 MathML Form

 Π ( n ; z m ) 2 Re ( z ) π Π ( n m ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 2 m sin ( z ) k = 0 j = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] n j m - j - k k ! ( 2 m sin 2 ( z ) i = 0 j + k - 1 ( - 1 2 ) j + k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "+", "k", "-", "i"]]], Pochhammer] m i sin 2 i ( z ) ( i + 1 ) ( j + k - i - 1 ) ! - ( 1 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] ( j + k ) ! ( log ( - m sin 2 ( z ) ) + ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 2 ) ) + ( 3 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] 2 m sin 2 ( z ) 3 F ~ 2 ( 1 , 1 , j + k + 3 2 ; j + k + 2 , 2 ; csc 2 ( z ) m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", "k", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "k", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["csc", "2"], "(", "z", ")"]], "m"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) Π ( n ; z m ) 2 Re ( z ) π Π ( n m ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 2 m sin ( z ) k = 0 j = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] n j m - j - k k ! ( 2 m sin 2 ( z ) i = 0 j + k - 1 ( - 1 2 ) j + k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "+", "k", "-", "i"]]], Pochhammer] m i sin 2 i ( z ) ( i + 1 ) ( j + k - i - 1 ) ! - ( 1 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] ( j + k ) ! ( log ( - m sin 2 ( z ) ) + ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 2 ) ) + ( 3 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] 2 m sin 2 ( z ) 3 F ~ 2 ( 1 , 1 , j + k + 3 2 ; j + k + 2 , 2 ; csc 2 ( z ) m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", "k", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "k", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["csc", "2"], "(", "z", ")"]], "m"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["n", "j"], " ", SuperscriptBox["m", RowBox[List[RowBox[List["-", "j"]], "-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["j", "+", "k"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "j", "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "j", "+", "k"]], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]], "]"]]]], RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["j", "+", "k"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", "k"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]]], "+", RowBox[List["2", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "j", "+", "k"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "i"]], "+", "j", "+", "k"]]]], "]"]], " ", SuperscriptBox["m", "i"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "i"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "i", "+", "j", "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]]]], RowBox[List["2", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]]]]

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

 2007-05-02