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 EllipticPi

 http://functions.wolfram.com/08.06.06.0095.01

 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]] + ((3 Csc[z]^2)/(8 m^2)) Sum[((Pochhammer[3/2, k] Pochhammer[5/2, k])/ (m^k (1 + k)!)) HypergeometricPFQRegularized[{1, 1, 5/2 + k}, {3 + k, 2}, Csc[z]^2/m], {k, 0, Infinity}] + Log[4] - ((2 Log[-4 m Sin[z]^2])/Pi) EllipticK[1/m] + (1/(4 m)) HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 1/m] + (9/(8 m^2)) Sum[(Pochhammer[5/2, k]^2/(m^k (2 + k)!^2)) Sum[1/(i + k + 3), {i, 0, k}], {k, 0, Infinity}] - (Sin[z]^2/2) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/ (m^k (k! (k + 1)!))) HypergeometricPFQ[{1, 1, 3/2 + k}, {2, 2 + k}, Sin[z]^2], {k, 0, Infinity}] + ((3 n)/(4 m^2 Sin[z]^2)) Sum[((n^j Pochhammer[5/2, k] Pochhammer[1/2, -j + k])/(m^k (-j + k)!)) HypergeometricPFQRegularized[{1, 1, 5/2 + k}, {3 + k, 2}, Csc[z]^2/m], {k, 0, Infinity}, {j, 0, k}] - ((n Log[-4 m Sin[z]^2])/(2 m)) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(m^k ((1 + k)! k!))) Hypergeometric2F1[1, -k, 1/2 - k, n], {k, 0, Infinity}] + (n/(2 m)) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(m^k (1 + k)!^2)) Hypergeometric2F1[1, -k, 1/2 - k, n], {k, 0, Infinity}] + ((9 n^2)/(8 m^2)) Sum[((n^k Pochhammer[5/2, k])/(m^k (2 + k)!)) Sum[1/(i + k + 3), {i, 0, k}], {k, 0, Infinity}] + ((3 n)/(4 m^2)) Sum[((n^j Pochhammer[3/2, -j + k] Pochhammer[5/2, k])/ (m^k ((-j + k + 1)! (k + 2)!))) Sum[1/(i + k + 3), {i, 0, k}], {k, 0, Infinity}, {j, 0, k}] - n Sin[z]^2 Sum[((Pochhammer[1/2, k] Pochhammer[1/2, j + k] Sin[z]^(2 j))/ (m^k ((1 + j) k! (j + k)!))) Hypergeometric2F1[1, -j - k, 1/2 - j - k, n], {k, 0, Infinity}, {j, 0, Infinity}]) + 2 Round[Re[z]/Pi] EllipticPi[n, m]

 Standard Form

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 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 ) k = 0 j = 0 m - k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] sin 2 j ( z ) ( j + 1 ) k ! ( j + k ) ! 2 F 1 ( 1 , - j - k ; - j - k + 1 2 ; n ) TagBox[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[RowBox[List["-", "j"]], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "k", "+", FractionBox["1", "2"]]], 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]], Hypergeometric2F1] - 2 log ( - 4 m sin 2 ( z ) ) π K ( 1 m ) + 1 4 m 4 F 3 ( 1 , 1 , 3 2 , 3 2 ; 2 , 2 , 2 ; 1 m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "m"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + log ( 4 ) - 2 log ( 1 - csc 2 ( z ) m + 1 ) - n log ( - 4 m sin 2 ( z ) ) 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 + 1 ) ! k ! 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[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]], Hypergeometric2F1] + 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 + 1 ) ! ) 2 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[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]], Hypergeometric2F1] - sin 2 ( z ) 2 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 ) ! 3 F 2 ( 1 , 1 , k + 3 2 ; 2 , k + 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["k", "+", 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["k", "+", "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] + 3 csc 2 ( z ) 8 m 2 k = 0 m - k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! 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] + 9 n 2 8 m 2 k = 0 m - k n k ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k + 2 ) ! i = 0 k 1 i + k + 3 + 9 8 m 2 k = 0 m - k ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 2 ( ( k + 2 ) ! ) 2 i = 0 k 1 i + k + 3 + 3 n 4 m 2 sin 2 ( z ) k = 0 j = 0 k m - k n j ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( k - j ) ! 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 n 4 m 2 k = 0 j = 0 k m - k n j ( 3 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k - j + 1 ) ! ( k + 2 ) ! i = 0 k 1 i + k + 3 ) 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 ) k = 0 j = 0 m - k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 2 ) j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["j", "+", "k"]]], Pochhammer] sin 2 j ( z ) ( j + 1 ) k ! ( j + k ) ! 2 F 1 ( 1 , - j - k ; - j - k + 1 2 ; n ) TagBox[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[RowBox[List["-", "j"]], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "j"]], "-", "k", "+", FractionBox["1", "2"]]], 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]], Hypergeometric2F1] - 2 log ( - 4 m sin 2 ( z ) ) π K ( 1 m ) + 1 4 m 4 F 3 ( 1 , 1 , 3 2 , 3 2 ; 2 , 2 , 2 ; 1 m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "m"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + log ( 4 ) - 2 log ( 1 - csc 2 ( z ) m + 1 ) - n log ( - 4 m sin 2 ( z ) ) 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 + 1 ) ! k ! 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[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]], Hypergeometric2F1] + 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 + 1 ) ! ) 2 2 F 1 ( 1 , - k ; 1 2 - k ; n ) TagBox[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]], Hypergeometric2F1] - sin 2 ( z ) 2 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 ) ! 3 F 2 ( 1 , 1 , k + 3 2 ; 2 , k + 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["k", "+", 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["k", "+", "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] + 3 csc 2 ( z ) 8 m 2 k = 0 m - k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k + 1 ) ! 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] + 9 n 2 8 m 2 k = 0 m - k n k ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k + 2 ) ! i = 0 k 1 i + k + 3 + 9 8 m 2 k = 0 m - k ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 2 ( ( k + 2 ) ! ) 2 i = 0 k 1 i + k + 3 + 3 n 4 m 2 sin 2 ( z ) k = 0 j = 0 k m - k n j ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( k - j ) ! 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 n 4 m 2 k = 0 j = 0 k m - k n j ( 3 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( k - j + 1 ) ! ( k + 2 ) ! i = 0 k 1 i + k + 3 ) [/itex]

 Rule Form

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

 2007-05-02