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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0009.01

 Input Form

 JacobiZeta[z, m] == JacobiZeta[Subscript[z, 0], m] + (Sqrt[1 - m Sin[Subscript[z, 0]]^2] - (EllipticE[m]/EllipticK[m]) (1/Sqrt[1 - m Sin[Subscript[z, 0]]^2])) (z - Subscript[z, 0]) + Sum[((I^(k - 1) Binomial[k - 3/2, k - 1])/k!) Sum[(((-1)^q Binomial[k - 1, q])/(1 - m Sin[Subscript[z, 0]]^2)^q) (Sqrt[1 - m Sin[Subscript[z, 0]]^2]/(1 - 2 q) + (EllipticE[m]/EllipticK[m]) ((1 - 2 k)/((1 + 2 q) Sqrt[1 - m Sin[Subscript[z, 0]]^2]))) Sum[Binomial[q, j] m^j (2 - m)^(q - j) 2^(k - j - q - 1) Sum[Binomial[j, i] (2 i - j)^(k - 1) E^(2 (2 i - j) I Subscript[z, 0]) (z - Subscript[z, 0])^k, {i, 0, j}], {j, 0, q}], {q, 1, k - 1}], {k, 2, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " "]], RowBox[List["EllipticK", "[", "m", "]"]]], FractionBox["1", RowBox[List[" ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]]]]]]]]]], ")"]], 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", "-", FractionBox["3", "2"]]], ",", RowBox[List["k", "-", "1"]]]], "]"]]]], RowBox[List["k", "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]]]], RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]], ")"]], "q"]]]], RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " "]], RowBox[List["EllipticK", "[", "m", "]"]]], FractionBox[RowBox[List["1", "-", RowBox[List["2", "k"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "q"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]]]]]]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox["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["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], " ", "\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]]]]

 MathML Form

 Ζ ( z m ) Ζ ( z 0 m ) + ( 1 - m sin 2 ( z 0 ) - E ( m ) K ( m ) 1 - m sin 2 ( z 0 ) ) ( z - z 0 ) + k = 2 k - 1 ( 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]] k ! q = 1 k - 1 ( - 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 sin 2 ( z 0 ) ) q ( E ( m ) ( 1 - 2 k ) K ( m ) ( ( 2 q + 1 ) 1 - m sin 2 ( z 0 ) ) + 1 - m sin 2 ( z 0 ) 1 - 2 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 2 ( 2 i - j ) z 0 ( z - z 0 ) k JacobiZeta z m JacobiZeta Subscript z 0 m 1 -1 m Subscript z 0 2 1 2 -1 EllipticE m EllipticK m 1 -1 m Subscript z 0 2 1 2 -1 z -1 Subscript z 0 k 2 k -1 Binomial k -1 3 2 k -1 k -1 q 1 k -1 -1 q Binomial k -1 q 1 -1 m Subscript z 0 2 q -1 EllipticE m 1 -1 2 k EllipticK m 2 q 1 1 -1 m Subscript z 0 2 1 2 -1 1 -1 m Subscript z 0 2 1 2 1 -1 2 q -1 j 0 q Binomial q j 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 2 2 i -1 j Subscript z 0 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]]]]]]], ")"]], " ", 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"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]], "+", FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "q"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]]]], ")"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox["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["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], " ", "\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]], ")"]], "q"]]]]]], RowBox[List["k", "!"]]]]]]]]]]]

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

 2007-05-02