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; }

 Sech

 http://functions.wolfram.com/01.24.21.0258.01

 Input Form

 Integrate[E^(p z) Cos[a z]^m Tanh[c z]^u Sech[c z], z] == (1/(p + c (1 + u))) (I^u 2^(1 - m) E^((p + c (u + 1)) z) Binomial[m, m/2] Binomial[u, u/2] HypergeometricPFQ[{1/2 + p/(2 c) + u/2, 1 + u}, {3/2 + p/(2 c) + u/2}, -E^(2 c z)] (1 - Mod[m, 2]) (1 - Mod[u, 2])) - I^u 2^(1 - m) E^(c (u + 1) z) Binomial[u, u/2] (-1 + Mod[u, 2]) Sum[Binomial[m, s] ((E^((p + I a (m - 2 s)) z) HypergeometricPFQ[ {1/2 + (I a m)/(2 c) + p/(2 c) - (I a s)/c + u/2, 1 + u}, {3/2 + (I a m)/(2 c) + p/(2 c) - (I a s)/c + u/2}, -E^(2 c z)])/ (p + I a (m - 2 s) + c (1 + u)) + (E^((p - I a (m - 2 s)) z) HypergeometricPFQ[{1/2 - (I a m)/(2 c) + p/(2 c) + (I a s)/c + u/2, 1 + u}, {3/2 - (I a m)/(2 c) + p/(2 c) + (I a s)/c + u/2}, -E^(2 c z)])/(p - I a (m - 2 s) + c (1 + u))), {s, 0, Floor[(1/2) (-1 + m)]}] + 2^(1 - m) E^(c (u + 1) z) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(-1)^k Binomial[u, k] ((1/(c (1 + 2 k) + p)) ((-1)^u E^((p - c (-2 k + u)) z) HypergeometricPFQ[{1/2 + k + p/(2 c), 1 + u}, {3/2 + k + p/(2 c)}, -E^(2 c z)]) + (E^((p + c (-2 k + u)) z) HypergeometricPFQ[ {1 + u, 1/2 - k + p/(2 c) + u}, {3/2 - k + p/(2 c) + u}, -E^(2 c z)])/(p + c (1 - 2 k + 2 u))), {k, 0, Floor[(1/2) (-1 + u)]}] + 2^(1 - m) E^(c (u + 1) z) Sum[(-1)^k Binomial[m, s] Binomial[u, k] (((-1)^u E^((p + I a (m - 2 s) - c (-2 k + u)) z) HypergeometricPFQ[{1/2 + k + (I a m)/(2 c) + p/(2 c) - (I a s)/c, 1 + u}, {3/2 + k + (I a m)/(2 c) + p/(2 c) - (I a s)/c}, -E^(2 c z)])/(c (1 + 2 k) + p + I a (m - 2 s)) + ((-1)^u E^((p - I a (m - 2 s) - c (-2 k + u)) z) HypergeometricPFQ[{1/2 + k - (I a m)/(2 c) + p/(2 c) + (I a s)/c, 1 + u}, {3/2 + k - (I a m)/(2 c) + p/(2 c) + (I a s)/c}, -E^(2 c z)])/(c (1 + 2 k) + p - I a (m - 2 s)) + (E^((p + I a (m - 2 s) + c (-2 k + u)) z) HypergeometricPFQ[ {1 + u, 1/2 - k + (I a m)/(2 c) + p/(2 c) - (I a s)/c + u}, {3/2 - k + (I a m)/(2 c) + p/(2 c) - (I a s)/c + u}, -E^(2 c z)])/ (p + I a (m - 2 s) + c (1 - 2 k + 2 u)) + (E^((p - I a (m - 2 s) + c (-2 k + u)) z) HypergeometricPFQ[ {1 + u, 1/2 - k - (I a m)/(2 c) + p/(2 c) + (I a s)/c + u}, {3/2 - k - (I a m)/(2 c) + p/(2 c) + (I a s)/c + u}, -E^(2 c z)])/ (p - I a (m - 2 s) + c (1 - 2 k + 2 u))), {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["\[ExponentialE]", RowBox[List["p", " ", "z"]]], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "u"], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", "u"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], ")"]], "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", 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["\[ImaginaryI]", "u"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], "z"]]], RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["(", 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[RowBox[List["(", RowBox[List["p", "+", 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["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], "z"]]], RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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[FractionBox["1", "2"], "+", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], "z"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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[FractionBox["1", "2"], "+", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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[FractionBox["1", "2"], "+", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]], "/;", 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

 p z cos m ( a z ) tanh u ( c z ) sech ( c z ) z 1 p + c ( u + 1 ) ( u 2 1 - m ( ( u + 1 ) c + p ) z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( p 2 c + u 2 + 1 2 , u + 1 ; p 2 c + u 2 + 3 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) ) - u 2 1 - m c ( u + 1 ) z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( u mod 2 \$CellContext`u 2 - 1 ) s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( p - a ( m - 2 s ) ) z 2 F 1 ( - a m 2 c + p 2 c + a s c + u 2 + 1 2 , u + 1 ; - a m 2 c + p 2 c + a s c + u 2 + 3 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( p - a ( m - 2 s ) + c ( u + 1 ) ) + ( ( p + a ( m - 2 s ) ) z 2 F 1 ( a m 2 c + p 2 c + u 2 + 1 2 - a s c , u + 1 ; a m 2 c + p 2 c + u 2 + 3 2 - a s c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"], "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"], "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( p + a ( m - 2 s ) + c ( u + 1 ) ) ) + 2 1 - m c ( u + 1 ) z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) k = 0 u - 1 2 ( - 1 ) k ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) u ( p - c ( u - 2 k ) ) z 2 F 1 ( k + p 2 c + 1 2 , u + 1 ; k + p 2 c + 3 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c ( 2 k + 1 ) + p + ( p + c ( u - 2 k ) ) z 2 F 1 ( u + 1 , - k + p 2 c + u + 1 2 ; - k + p 2 c + u + 3 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u", "+", FractionBox["1", "2"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u", "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] p + c ( - 2 k + 2 u + 1 ) ) + 2 1 - m c ( u + 1 ) z k = 0 u - 1 2 s = 0 m - 1 2 ( - 1 ) k ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( - 1 ) u ( p - a ( m - 2 s ) - c ( u - 2 k ) ) z 2 F 1 ( k + p 2 c + a s c + 1 2 - a m 2 c , u + 1 ; k + p 2 c + a s c + 3 2 - a m 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( c ( 2 k + 1 ) + p - a ( m - 2 s ) ) + ( ( - 1 ) u ( p + a ( m - 2 s ) - c ( u - 2 k ) ) z 2 F 1 ( k + a m 2 c + p 2 c + 1 2 - a s c , u + 1 ; k + a m 2 c + p 2 c + 3 2 - a s c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( c ( 2 k + 1 ) + p + a ( m - 2 s ) ) + ( ( p - a ( m - 2 s ) + c ( u - 2 k ) ) z 2 F 1 ( u + 1 , - k + p 2 c + a s c + u + 1 2 - a m 2 c ; - k + p 2 c + a s c + u + 3 2 - a m 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u", "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( p - a ( m - 2 s ) + c ( - 2 k + 2 u + 1 ) ) + ( ( p + a ( m - 2 s ) + c ( u - 2 k ) ) z 2 F 1 ( u + 1 , - k + a m 2 c + p 2 c + u + 1 2 - a s c ; - k + a m 2 c + p 2 c + u + 3 2 - a s c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u", "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u", "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( p + a ( m - 2 s ) + c ( - 2 k + 2 u + 1 ) ) ) /; m + u + Condition z p z a z m c z u c z 1 p c u 1 -1 u 2 1 -1 m u 1 c p z Binomial m m 2 -1 Binomial u u 2 -1 HypergeometricPFQ p 2 c -1 u 2 -1 1 2 u 1 p 2 c -1 u 2 -1 3 2 -1 2 c z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 -1 u 2 1 -1 m c u 1 z Binomial u u 2 -1 \$CellContext`u 2 -1 s 0 m -1 2 -1 Binomial m s p -1 a m -1 2 s z HypergeometricPFQ -1 a m 2 c -1 p 2 c -1 a s c -1 u 2 -1 1 2 u 1 -1 a m 2 c -1 p 2 c -1 a s c -1 u 2 -1 3 2 -1 2 c z p -1 a m -1 2 s c u 1 -1 p a m -1 2 s z HypergeometricPFQ a m 2 c -1 p 2 c -1 u 2 -1 1 2 -1 a s c -1 u 1 a m 2 c -1 p 2 c -1 u 2 -1 3 2 -1 a s c -1 -1 2 c z p a m -1 2 s c u 1 -1 2 1 -1 m c u 1 z Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 u -1 2 -1 -1 k Binomial u k -1 u p -1 c u -1 2 k z HypergeometricPFQ k p 2 c -1 1 2 u 1 k p 2 c -1 3 2 -1 2 c z c 2 k 1 p -1 p c u -1 2 k z HypergeometricPFQ u 1 -1 k p 2 c -1 u 1 2 -1 k p 2 c -1 u 3 2 -1 2 c z p c -2 k 2 u 1 -1 2 1 -1 m c u 1 z s 0 m -1 2 -1 k 0 u -1 2 -1 -1 k Binomial m s Binomial u k -1 u p -1 a m -1 2 s -1 c u -1 2 k z HypergeometricPFQ k p 2 c -1 a s c -1 1 2 -1 a m 2 c -1 u 1 k p 2 c -1 a s c -1 3 2 -1 a m 2 c -1 -1 2 c z c 2 k 1 p -1 a m -1 2 s -1 -1 u p a m -1 2 s -1 c u -1 2 k z HypergeometricPFQ k a m 2 c -1 p 2 c -1 1 2 -1 a s c -1 u 1 k a m 2 c -1 p 2 c -1 3 2 -1 a s c -1 -1 2 c z c 2 k 1 p a m -1 2 s -1 p -1 a m -1 2 s c u -1 2 k z HypergeometricPFQ u 1 -1 k p 2 c -1 a s c -1 u 1 2 -1 a m 2 c -1 -1 k p 2 c -1 a s c -1 u 3 2 -1 a m 2 c -1 -1 2 c z p -1 a m -1 2 s c -2 k 2 u 1 -1 p a m -1 2 s c u -1 2 k z HypergeometricPFQ u 1 -1 k a m 2 c -1 p 2 c -1 u 1 2 -1 a s c -1 -1 k a m 2 c -1 p 2 c -1 u 3 2 -1 a s c -1 -1 2 c z p a m -1 2 s c -2 k 2 u 1 -1 m SuperPlus u SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "u_"], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "u"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", 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["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", "u"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["(", 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[RowBox[List["(", RowBox[List["p", "+", 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["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["u", "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "u"]], ")"]]]]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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[FractionBox["1", "2"], "+", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], " ", "z"]]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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[FractionBox["1", "2"], "+", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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[FractionBox["1", "2"], "+", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], ",", RowBox[List["1", "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", 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["1", "+", "u"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", "k", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", "u"]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], ")"]]]]]]]]]]]], "/;", 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