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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0014.01

 Input Form

 JacobiZeta[z, m] == Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] JacobiZeta[x, m] + JacobiZeta[ArcCsc[Sqrt[m]], m] (1 - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] Sum[(1/k!) Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[x]^q (2 p + q - j)^k Binomial[j - q, p] Sum[((Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)! (2 Sin[x])^ (j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/2, s] m^(i - s) Cos[x]^(-1 - 2 s) (1 - m Sin[x]^2)^(-i + s) (Pochhammer[-(1/2), i - s] Sqrt[1 - m Sin[x]^2] - (EllipticE[m]/EllipticK[m]) (Pochhammer[1/2, i - s]/ Sqrt[1 - m Sin[x]^2])), {s, 0, i}]) (z - x)^k, {i, 0, j - 1}])/E^((1/2) I ((k - 2 p - q + j) Pi + 2 (2 p + q - j) x)), {p, 0, j - q}], {q, 0, j - 1}], {j, 1, k}], {k, 1, Infinity}] /; Element[x, Reals] && Element[m, Reals] && m > 1 && ArcCsc[Sqrt[m]] + Pi u < x < Pi/2 + Pi u && Element[u, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["JacobiZeta", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "x"]]]], ")"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "x", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", "s"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", "i"]], "+", "s"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " "]], RowBox[List["EllipticK", "[", "m", "]"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["i", "-", "s"]]]], "]"]], " "]], SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]]]]], ")"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["JacobiZeta", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], " ", "x"]]]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "s"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", "i"]], "+", "s"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["i", "-", "s"]]]], "]"]]]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]]]]], ")"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "x", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]]]]]]]]]]]], RowBox[List["j", "!"]]]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02