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/08.06.06.0096.01

 Input Form

 EllipticPi[n, z, m] == (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/ (2 m Sin[z])) (-2 Log[1 + Sqrt[1 - Csc[z]^2/m]] - Log[(-m) Sin[z]^2] + (1/(2 m)) Sum[(Pochhammer[3/2, k]/(m^k (1 + k)!)) (((3 Csc[z]^2 Pochhammer[5/2, k])/(4 m)) HypergeometricPFQRegularized[ {1, 1, 5/2 + k}, {3 + k, 2}, Csc[z]^2/m] - (Pochhammer[3/2, k]/(2 (1 + k)!)) (Log[-4 m Sin[z]^2] - 1/(k + 1) - Sum[2/(i + k + 2), {i, 0, k - 1}])), {k, 0, Infinity}] - (Sin[z]^2/2) Sum[((Pochhammer[3/2, i] Pochhammer[1/2, i])/ (m^i (i! (i + 1)!))) HypergeometricPFQ[{1, 1, 3/2 + i}, {2, 2 + i}, Sin[z]^2], {i, 0, Infinity}] + (n/m) Sum[(((3 Csc[z]^2 Pochhammer[5/2, k])/(4 m)) HypergeometricPFQRegularized[{1, 1, 5/2 + k}, {3 + k, 2}, Csc[z]^2/m] + (Pochhammer[3/2, k]/(1 + k)!) Sum[1/(2 + i + k), {i, 0, -1 + k}]) (n/m)^k Sum[Pochhammer[1/2, j]/(n^j j!), {j, 0, k}], {k, 0, Infinity}] + (n/(2 m)) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/ (m^k (k! (1 + k)!))) (1/(1 + k) - Log[-4 m Sin[z]^2]) Hypergeometric2F1[1, -k, 1/2 - k, n], {k, 0, Infinity}] - n Sin[z]^2 Sum[(Pochhammer[1/2, i]/i!) (n/m)^i Sum[(Pochhammer[1/2, j]/(n^j j!)) HypergeometricPFQ[{1, 1, 1/2 + j}, {2, 1 + j}, Sin[z]^2], {j, 0, i}], {i, 0, Infinity}] - ((n^2 Sin[z]^4)/2) Sum[(Pochhammer[1/2, i]/i!) AppellF1[2, 1/2, 1, 3, Sin[z]^2, n Sin[z]^2] (n/m)^i, {i, 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["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", 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RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "-", FractionBox["1", RowBox[List["k", "+", "1"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox["2", RowBox[List["i", "+", "k", "+", "2"]]]]]]], ")"]]]]]], ")"]]]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["-", "i"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "i"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " "]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "+", "1"]], ")"]], "!"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "i"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "i"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]]]]]], " ", "+", RowBox[List[FractionBox["n", "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["3", " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "k"]], "]"]]]], RowBox[List["4", " ", "m"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["5", "2"], "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["3", "+", "k"]], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], "m"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], FractionBox["1", RowBox[List["2", "+", "i", "+", "k"]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["n", "m"], ")"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox["n", RowBox[List["-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " "]], RowBox[List["j", "!"]]]]]]]]]]], "+", RowBox[List[FractionBox["n", RowBox[List["2", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", "k"]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], ")"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "n"]], "]"]]]]]]]], " ", "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], RowBox[List["i", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["n", "m"], ")"]], "i"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["n", RowBox[List["-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " "]]]], RowBox[List[" ", RowBox[List["j", "!"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["1", "2"], "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "+", "j"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["n", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "4"]]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " "]], RowBox[List["i", "!"]]], RowBox[List["AppellF1", "[", RowBox[List["2", ",", FractionBox["1", "2"], ",", "1", ",", "3", ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox["n", "m"], ")"]], "i"]]]]]]]]], ")"]]]], "+", 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 ) ( - n sin 2 ( z ) i = 0 ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! ( n m ) i j = 0 i n - j ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] j ! 3 F 2 ( 1 , 1 , j + 1 2 ; 2 , j + 1 ; sin 2 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["sin", "2"], "(", "z", ")"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 log ( 1 - csc 2 ( z ) m + 1 ) - log ( - m sin 2 ( z ) ) - n 2 sin 4 ( z ) 2 i = 0 ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! F 1 AppellF1 ( 2 ; 1 2 , 1 ; 3 ; sin 2 ( z ) , n sin 2 ( z ) ) ( n m ) i + n 2 m k = 0 m - k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( k + 1 ) ! ( 1 k + 1 - log ( - 4 m sin 2 ( z ) ) ) 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["n", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] - sin 2 ( z ) 2 i = 0 m - i ( 3 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "i"], Pochhammer] ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! ( i + 1 ) ! 3 F 2 ( 1 , 1 , i + 3 2 ; 2 , i + 2 ; sin 2 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["i", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["i", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["sin", "2"], "(", "z", ")"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 2 m k = 0 m - k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! ( 3 csc 2 ( z ) ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 4 m 3 F ~ 2 ( 1 , 1 , k + 5 2 ; k + 3 , 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["k", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], 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] - ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 ( k + 1 ) ! ( log ( - 4 m sin 2 ( z ) ) - i = 0 k - 1 2 i + k + 2 - 1 k + 1 ) ) + n m k = 0 ( 3 csc 2 ( z ) ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 4 m 3 F ~ 2 ( 1 , 1 , k + 5 2 ; k + 3 , 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["k", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], 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] + ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! i = 0 k - 1 1 i + k + 2 ) ( n m ) k j = 0 k n - j j ! ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] ) Π ( n ; z m ) 2 Re ( z ) π Π ( n m ) + ( - 1 ) Re ( z ) π - m sin 2 ( z ) 2 m sin ( z ) ( - n sin 2 ( z ) i = 0 ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! ( n m ) i j = 0 i n - j ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] j ! 3 F 2 ( 1 , 1 , j + 1 2 ; 2 , j + 1 ; sin 2 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["sin", "2"], "(", "z", ")"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 log ( 1 - csc 2 ( z ) m + 1 ) - log ( - m sin 2 ( z ) ) - n 2 sin 4 ( z ) 2 i = 0 ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! F 1 AppellF1 ( 2 ; 1 2 , 1 ; 3 ; sin 2 ( z ) , n sin 2 ( z ) ) ( n m ) i + n 2 m k = 0 m - k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( k + 1 ) ! ( 1 k + 1 - log ( - 4 m sin 2 ( z ) ) ) 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["n", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] - sin 2 ( z ) 2 i = 0 m - i ( 3 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "i"], Pochhammer] ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] i ! ( i + 1 ) ! 3 F 2 ( 1 , 1 , i + 3 2 ; 2 , i + 2 ; sin 2 ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["i", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["i", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["sin", "2"], "(", "z", ")"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 2 m k = 0 m - k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! ( 3 csc 2 ( z ) ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 4 m 3 F ~ 2 ( 1 , 1 , k + 5 2 ; k + 3 , 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["k", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], 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] - ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] 2 ( k + 1 ) ! ( log ( - 4 m sin 2 ( z ) ) - i = 0 k - 1 2 i + k + 2 - 1 k + 1 ) ) + n m k = 0 ( 3 csc 2 ( z ) ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 4 m 3 F ~ 2 ( 1 , 1 , k + 5 2 ; k + 3 , 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["k", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", "3"]], 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] + ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! i = 0 k - 1 1 i + k + 2 ) ( n m ) k j = 0 k n - j j ! ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] ) [/itex]

 Rule Form

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RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", RowBox[List["-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", "k"]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "n"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], RowBox[List["2", " ", "m"]]], "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], 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RowBox[List[SuperscriptBox["n", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "4"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List["2", ",", FractionBox["1", "2"], ",", "1", ",", "3", ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["n", "m"], ")"]], "i"]]], RowBox[List["i", "!"]]]]]]]]], ")"]]]], 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