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; }

 Cosh

 http://functions.wolfram.com/01.20.21.3666.01

 Input Form

 Integrate[z^n Cos[b z^2]^m Cosh[f z + g]^v, z] == (1/(1 + n)) (2^(-m - v) z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) - 2^(-1 - m - v) z^(1 + n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, k] (((-I) b (-2 k + m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-I) b (-2 k + m) z^2] + (I b (-2 k + m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, I b (-2 k + m) z^2]), {k, 0, Floor[(1/2) (-1 + m)]}] + (-1)^n 2^(-m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, k] (E^(g (-2 k + v)) (f (-2 k + v))^(-1 - n) Gamma[1 + n, (-f) (-2 k + v) z] + (((-f) (-2 k + v))^(-1 - n) Gamma[1 + n, f (-2 k + v) z])/E^(g (-2 k + v))), {k, 0, Floor[(1/2) (-1 + v)]}] - 2^(-1 - m - v) Sum[Binomial[v, k] Sum[Binomial[m, s] (E^((-g) (-2 k + v) - (I f^2 (-2 k + v)^2)/(4 b (m - 2 s))) ((-I) b (m - 2 s))^(-1 - n) Sum[2^(j - n) (f (-2 k + v))^(-j + n) ((-f) (-2 k + v) - 2 I b (m - 2 s) z)^(1 + j) (-((I ((-f) (-2 k + v) - 2 I b (m - 2 s) z)^2)/(b (m - 2 s))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((I ((-f) (-2 k + v) - 2 I b (m - 2 s) z)^2)/(4 b (m - 2 s)))], {j, 0, n}] + E^(g (-2 k + v) - (I f^2 (-2 k + v)^2)/ (4 b (m - 2 s))) ((-I) b (m - 2 s))^(-1 - n) Sum[2^(j - n) ((-f) (-2 k + v))^(-j + n) (f (-2 k + v) - 2 I b (m - 2 s) z)^(1 + j) (-((I (f (-2 k + v) - 2 I b (m - 2 s) z)^2)/(b (m - 2 s))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((I (f (-2 k + v) - 2 I b (m - 2 s) z)^2)/(4 b (m - 2 s)))], {j, 0, n}] + E^((-g) (-2 k + v) + (I f^2 (-2 k + v)^2)/ (4 b (m - 2 s))) (I b (m - 2 s))^(-1 - n) Sum[2^(j - n) (f (-2 k + v))^(-j + n) ((-f) (-2 k + v) + 2 I b (m - 2 s) z)^(1 + j) ((I ((-f) (-2 k + v) + 2 I b (m - 2 s) z)^2)/(b (m - 2 s)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, (I ((-f) (-2 k + v) + 2 I b (m - 2 s) z)^2)/(4 b (m - 2 s))], {j, 0, n}] + E^(g (-2 k + v) + (I f^2 (-2 k + v)^2)/(4 b (m - 2 s))) (I b (m - 2 s))^(-1 - n) Sum[2^(j - n) ((-f) (-2 k + v))^(-j + n) (f (-2 k + v) + 2 I b (m - 2 s) z)^(1 + j) ((I (f (-2 k + v) + 2 I b (m - 2 s) z)^2)/(b (m - 2 s)))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, (I (f (-2 k + v) + 2 I b (m - 2 s) z)^2)/(4 b (m - 2 s))], {j, 0, n}]), {s, 0, Floor[(1/2) (-1 + m)]}], {k, 0, Floor[(1/2) (-1 + v)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "+", "n"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "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", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", 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"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "k"]], "]"]], " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 z n cos m ( b z 2 ) cosh v ( g + f z ) z - 2 - m - v - 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( n + 1 2 , - b ( m - 2 k ) z 2 ) ( - b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) + ( b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , b ( m - 2 k ) z 2 ) ) ) z n + 1 + 2 - m - v z n + 1 ( 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]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) n + 1 + ( - 1 ) n 2 - m - v ( 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 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - g ( v - 2 k ) Γ ( n + 1 , f ( v - 2 k ) z ) ( - f ( v - 2 k ) ) - n - 1 + g ( v - 2 k ) ( f ( v - 2 k ) ) - n - 1 Γ ( n + 1 , - f ( v - 2 k ) z ) ) - 2 - m - v - 1 k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 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]]]]] ( - f 2 ( v - 2 k ) 2 4 b ( m - 2 s ) - g ( v - 2 k ) ( j = 0 n 2 j - n ( f ( v - 2 k ) ) n - j ( - f ( v - 2 k ) - 2 b ( m - 2 s ) z ) j + 1 ( - ( - f ( v - 2 k ) - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( - f ( v - 2 k ) - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) ( - b ( m - 2 s ) ) - n - 1 + - f 2 ( v - 2 k ) 2 4 b ( m - 2 s ) + g ( v - 2 k ) ( j = 0 n 2 j - n ( - f ( v - 2 k ) ) n - j ( f ( v - 2 k ) - 2 b ( m - 2 s ) z ) j + 1 ( - ( f ( v - 2 k ) - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( f ( v - 2 k ) - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) ( - b ( m - 2 s ) ) - n - 1 + f 2 ( v - 2 k ) 2 4 b ( m - 2 s ) - g ( v - 2 k ) ( b ( m - 2 s ) ) - n - 1 j = 0 n 2 j - n ( f ( v - 2 k ) ) n - j ( 2 b ( m - 2 s ) z - f ( v - 2 k ) ) j + 1 ( ( 2 b ( m - 2 s ) z - f ( v - 2 k ) ) 2 b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , ( 2 b ( m - 2 s ) z - f ( v - 2 k ) ) 2 4 b ( m - 2 s ) ) + f 2 ( v - 2 k ) 2 4 b ( m - 2 s ) + g ( v - 2 k ) ( b ( m - 2 s ) ) - n - 1 j = 0 n 2 j - n ( - f ( v - 2 k ) ) n - j ( f ( v - 2 k ) + 2 b ( m - 2 s ) z ) j + 1 ( ( f ( v - 2 k ) + 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , ( f ( v - 2 k ) + 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) /; n m + v + Condition z z n b z 2 m g f z v -1 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 Binomial m k Gamma n 1 2 -1 -1 b m -1 2 k z 2 -1 b m -1 2 k z 2 1 2 -1 n -1 b m -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 b m -1 2 k z 2 z n 1 2 -1 m -1 v z n 1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 n 1 -1 -1 n 2 -1 m -1 v Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 Binomial v k -1 g v -1 2 k Gamma n 1 f v -1 2 k z -1 f v -1 2 k -1 n -1 g v -1 2 k f v -1 2 k -1 n -1 Gamma n 1 -1 f v -1 2 k z -1 2 -1 m -1 v -1 k 0 v -1 2 -1 Binomial v k s 0 m -1 2 -1 Binomial m s -1 f 2 v -1 2 k 2 4 b m -1 2 s -1 -1 g v -1 2 k j 0 n 2 j -1 n f v -1 2 k n -1 j -1 f v -1 2 k -1 2 b m -1 2 s z j 1 -1 -1 f v -1 2 k -1 2 b m -1 2 s z 2 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 -1 f v -1 2 k -1 2 b m -1 2 s z 2 4 b m -1 2 s -1 -1 b m -1 2 s -1 n -1 -1 f 2 v -1 2 k 2 4 b m -1 2 s -1 g v -1 2 k j 0 n 2 j -1 n -1 f v -1 2 k n -1 j f v -1 2 k -1 2 b m -1 2 s z j 1 -1 f v -1 2 k -1 2 b m -1 2 s z 2 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f v -1 2 k -1 2 b m -1 2 s z 2 4 b m -1 2 s -1 -1 b m -1 2 s -1 n -1 f 2 v -1 2 k 2 4 b m -1 2 s -1 -1 g v -1 2 k b m -1 2 s -1 n -1 j 0 n 2 j -1 n f v -1 2 k n -1 j 2 b m -1 2 s z -1 f v -1 2 k j 1 2 b m -1 2 s z -1 f v -1 2 k 2 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 2 b m -1 2 s z -1 f v -1 2 k 2 4 b m -1 2 s -1 f 2 v -1 2 k 2 4 b m -1 2 s -1 g v -1 2 k b m -1 2 s -1 n -1 j 0 n 2 j -1 n -1 f v -1 2 k n -1 j f v -1 2 k 2 b m -1 2 s z j 1 f v -1 2 k 2 b m -1 2 s z 2 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 f v -1 2 k 2 b m -1 2 s z 2 4 b m -1 2 s -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["1", "+", "n"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "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", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", 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"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "k"]], "]"]], " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "g"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18