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; }

 Csch

 http://functions.wolfram.com/01.23.21.0222.01

 Input Form

 Integrate[Cos[a z]^m Coth[c z]^u Csch[c z], z] == (-(1/(c (u + 1)))) (2^(1 - m - u) E^(c z) (1 - E^(2 c z))^u Binomial[m, m/2] Binomial[u, u/2] Csch[c z]^u HypergeometricPFQ[{(u + 1)/2, u + 1}, {(u + 3)/2}, E^(2 c z)] (1 - Mod[m, 2]) (1 - Mod[u, 2])) - 2^(1 - m - u) E^(c z) (1 - E^(2 c z))^u Binomial[u, u/2] Csch[c z]^u (1 - Mod[u, 2]) Sum[Binomial[m, s] ((E^(I a (m - 2 s) z) HypergeometricPFQ[ {(I a m)/(2 c) - (I a s)/c + u/2 + 1/2, u + 1}, {(I a m)/(2 c) - (I a s)/c + u/2 + 3/2}, E^(2 c z)])/ (I a (m - 2 s) + c (u + 1)) + HypergeometricPFQ[{-((I a m)/(2 c)) + (I a s)/c + u/2 + 1/2, u + 1}, {-((I a m)/(2 c)) + (I a s)/c + u/2 + 3/2}, E^(2 c z)]/ E^(I a (m - 2 s) z)/((-I) a (m - 2 s) + c (u + 1))), {s, 0, Floor[(1/2) (-1 + m)]}] - 2^(1 - m - u) E^(c z) (1 - E^(2 c z))^u Binomial[m, m/2] Csch[c z]^u (1 - Mod[m, 2]) Sum[Binomial[u, k] (HypergeometricPFQ[{k + 1/2, u + 1}, {k + 3/2}, E^(2 c z)]/(E^(c (-2 k + u) z) (c (2 k + 1))) + (E^(c (-2 k + u) z) HypergeometricPFQ[{-k + u + 1/2, u + 1}, {-k + u + 3/2}, E^(2 c z)])/(c (2 u - 2 k + 1))), {k, 0, Floor[(1/2) (-1 + u)]}] - 2^(1 - m - u) E^(c z) (1 - E^(2 c z))^u Csch[c z]^u Sum[Binomial[m, s] Binomial[u, k] ((E^((I a (m - 2 s) - c (-2 k + u)) z) HypergeometricPFQ[ {k + (I a m)/(2 c) - (I a s)/c + 1/2, u + 1}, {k + (I a m)/(2 c) - (I a s)/c + 3/2}, E^(2 c z)])/ (I a (m - 2 s) + c (2 k + 1)) + (E^(((-I) a (m - 2 s) - c (-2 k + u)) z) HypergeometricPFQ[ {k - (I a m)/(2 c) + (I a s)/c + 1/2, u + 1}, {k - (I a m)/(2 c) + (I a s)/c + 3/2}, E^(2 c z)])/ ((-I) a (m - 2 s) + c (2 k + 1)) + (E^((I a (m - 2 s) + c (-2 k + u)) z) HypergeometricPFQ[ {-k + (I a m)/(2 c) - (I a s)/c + u + 1/2, u + 1}, {-k + (I a m)/(2 c) - (I a s)/c + u + 3/2}, E^(2 c z)])/ (I a (m - 2 s) + c (2 u - 2 k + 1)) + (E^(((-I) a (m - 2 s) + c (-2 k + u)) z) HypergeometricPFQ[ {-k - (I a m)/(2 c) + (I a s)/c + u + 1/2, u + 1}, {-k - (I a m)/(2 c) + (I a s)/c + u + 3/2}, E^(2 c z)])/ ((-I) a (m - 2 s) + c (2 u - 2 k + 1))), {k, 0, Floor[(1/2) (-1 + u)]}, {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["u", "+", "3"]], "2"], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "k"]], "+", "1"]], ")"]]]], ")"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "k"]], "+", "1"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "u"]], "-", RowBox[List["2", "k"]], "+", "1"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["u", ">", "0"]]]]]]]]

 MathML Form

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "u_"], " ", RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["u", "+", "1"]], "2"], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["u", "+", "3"]], "2"], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "u"]], "-", RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "u"], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "u"]], "-", RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["1", "2"]]], ",", RowBox[List["u", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "k"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["3", "2"]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "u"]], "-", RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["u", ">", "0"]]]]]]]]]]

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

 2002-12-18