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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0029.01

 Input Form

 JacobiZeta[z, m] == (-JacobiZeta[ArcCsc[Sqrt[m]], m]) (I Sqrt[-(1/(z - Subscript[z, 0])^2)] (z - Subscript[z, 0]) + Sqrt[(z - Subscript[z, 0])^2]/(z - Subscript[z, 0])) + (Sqrt[1 - m] - EllipticE[m]/(Sqrt[1 - m] EllipticK[m])) (z - Subscript[z, 0]) + Sum[((I^(k - 1) Binomial[k - 3/2, k - 1])/k!) Sum[((Sqrt[1 - m]/(1 - 2 q) - ((2 k - 1) EllipticE[m])/ (Sqrt[1 - m] (2 q + 1) EllipticK[m])) (-1)^q Binomial[k - 1, q] Sum[Binomial[q, j] (-m)^j (2 - m)^(q - j) 2^(k - j - q - 1) Sum[Binomial[j, i] (2 i - j)^(k - 1) (z - Subscript[z, 0])^k, {i, 0, j}], {j, 0, q}])/(1 - m)^q, {q, 1, k - 1}], {k, 2, Infinity}] /; (z -> Subscript[z, 0]) && Subscript[z, 0] == Pi/2 + Pi u && Element[u, Integers] && Element[m, Reals] && m > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", FractionBox[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]], RowBox[List["z", "-", SubscriptBox["z", "0"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", RowBox[List["3", "/", "2"]]]], ",", RowBox[List["k", "-", "1"]]]], "]"]], " "]], RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["1", "-", RowBox[List["2", "q"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]], ")"]], RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["-", "q"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], " ", ")"]], RowBox[List["q", "-", "j"]]], SuperscriptBox["2", RowBox[List["k", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["k", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]]]]]]]]

 MathML Form

 Ζ ( z m ) - Ζ ( csc - 1 ( m ) m ) ( - 1 ( z - z 0 ) 2 ( z - z 0 ) + ( z - z 0 ) 2 z - z 0 ) + ( 1 - m - E ( m ) 1 - m K ( m ) ) ( z - z 0 ) + k = 2 k - 1 k ! ( k - 3 2 k - 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", FractionBox["3", "2"]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] q = 1 k - 1 ( 1 - m 1 - 2 q - ( 2 k - 1 ) E ( m ) 1 - m ( 2 q + 1 ) K ( m ) ) ( - 1 ) q ( k - 1 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m ) - q j = 0 q ( q j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["q", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - m ) j ( 2 - m ) q - j 2 - j + k - q - 1 i = 0 j ( j i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 i - j ) k - 1 ( z - z 0 ) k /; ( z "\[Rule]" z 0 ) z 0 π u + π 2 u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 Condition JacobiZeta z m -1 JacobiZeta m 1 2 m -1 1 z -1 Subscript z 0 2 -1 1 2 z -1 Subscript z 0 z -1 Subscript z 0 2 1 2 z -1 Subscript z 0 -1 1 -1 m 1 2 -1 EllipticE m 1 -1 m 1 2 EllipticK m -1 z -1 Subscript z 0 k 2 k -1 k -1 Binomial k -1 3 2 k -1 q 1 k -1 1 -1 m 1 2 1 -1 2 q -1 -1 2 k -1 EllipticE m 1 -1 m 1 2 2 q 1 EllipticK m -1 -1 q Binomial k -1 q 1 -1 m -1 q j 0 q Binomial q j -1 m j 2 -1 m q -1 j 2 -1 j k -1 q -1 i 0 j Binomial j i 2 i -1 j k -1 z -1 Subscript z 0 k Rule z Subscript z 0 Subscript z 0 u 2 -1 u m m 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", FractionBox[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]], RowBox[List["z", "-", SubscriptBox["zz", "0"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", FractionBox["3", "2"]]], ",", RowBox[List["k", "-", "1"]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["-", "q"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]], RowBox[List["q", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List["k", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]]]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]]]]]]]]]]

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

 2007-05-02