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/01.20.21.3667.01

 Input Form

 Integrate[z^n Cos[b Sqrt[z]]^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])) + (-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)]}] + (-1)^n 2^(1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(Binomial[m, k] (Gamma[2 (1 + n), (-I) b (-2 k + m) Sqrt[z]] + Gamma[2 (1 + n), I b (-2 k + m) Sqrt[z]]))/ (b (-2 k + m))^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - 2 n - v) Sum[Binomial[v, k] Sum[Binomial[m, s] ((E^(-((b^2 (m - 2 s)^2)/(4 f (-2 k + v))) - g (-2 k + v)) Sum[(-1)^(-h + j) 4^j ((-I) b (m - 2 s))^ (-h - j + 2 n) ((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^ (h + j) (((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v)))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-I) b (m - 2 s) ((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), ((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/ (4 f (-2 k + v))] - 2 f (-2 k + v) Sqrt[ ((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v))] Gamma[(1/2) (2 + h + j), ((-I) b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v))]), {j, 0, n}, {h, 0, j}])/ ((-f) (-2 k + v))^(2 (1 + n)) + (E^(-((b^2 (m - 2 s)^2)/(4 f (-2 k + v))) - g (-2 k + v)) Sum[(-1)^(-h + j) 4^j (I b (m - 2 s))^(-h - j + 2 n) (I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^(h + j) ((I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v)))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (I b (m - 2 s) (I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z]) Gamma[ (1/2) (1 + h + j), (I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/ (4 f (-2 k + v))] - 2 f (-2 k + v) Sqrt[ (I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v))] Gamma[(1/2) (2 + h + j), (I b (m - 2 s) - 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v))]), {j, 0, n}, {h, 0, j}])/ ((-f) (-2 k + v))^(2 (1 + n)) + (E^((b^2 (m - 2 s)^2)/(4 f (-2 k + v)) + g (-2 k + v)) Sum[(-1)^(-h + j) 4^j ((-I) b (m - 2 s))^(-h - j + 2 n) ((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^(h + j) (-(((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/ (f (-2 k + v))))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-I) b (m - 2 s) ((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/ (4 f (-2 k + v)))] + 2 f (-2 k + v) Sqrt[ -(((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/ (f (-2 k + v)))] Gamma[(1/2) (2 + h + j), -(((-I) b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/ (4 f (-2 k + v)))]), {j, 0, n}, {h, 0, j}])/ (f (-2 k + v))^(2 (1 + n)) + (E^((b^2 (m - 2 s)^2)/(4 f (-2 k + v)) + g (-2 k + v)) Sum[(-1)^(-h + j) 4^j (I b (m - 2 s))^(-h - j + 2 n) (I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^(h + j) (-((I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v))))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (I b (m - 2 s) (I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z]) Gamma[ (1/2) (1 + h + j), -((I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^ 2/(4 f (-2 k + v)))] + 2 f (-2 k + v) Sqrt[ -((I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/(f (-2 k + v)))] Gamma[(1/2) (2 + h + j), -((I b (m - 2 s) + 2 f (-2 k + v) Sqrt[z])^2/(4 f (-2 k + v)))]), {j, 0, n}, {h, 0, j}])/ (f (-2 k + v))^(2 (1 + 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", " ", SqrtBox["z"]]], "]"]], "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[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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "v"]]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", RowBox[List["2", " ", "n"]], "-", "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["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "-", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "-", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], "+", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], "+", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]]]]]], "/;", 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 ) cosh v ( g + f z ) z 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 ) ) + ( - 1 ) n 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 ( b ( m - 2 k ) ) - 2 ( n + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( 2 ( n + 1 ) , - b ( m - 2 k ) z ) + Γ ( 2 ( n + 1 ) , b ( m - 2 k ) z ) ) + 2 - m - 2 n - 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]]]]] ( - b 2 ( m - 2 s ) 2 4 f ( v - 2 k ) - g ( v - 2 k ) ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ( m - 2 s ) ) - h - j + 2 n ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) h + j ( ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - b ( m - 2 s ) ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) - 2 f ( v - 2 k ) ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( - b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) ) ) ( - f ( v - 2 k ) ) - 2 ( n + 1 ) + - b 2 ( m - 2 s ) 2 4 f ( v - 2 k ) - g ( v - 2 k ) ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ( m - 2 s ) ) - h - j + 2 n ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) h + j ( ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 s ) ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) - 2 f ( v - 2 k ) ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( b ( m - 2 s ) - 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) ) ) ( - f ( v - 2 k ) ) - 2 ( n + 1 ) + b 2 ( m - 2 s ) 2 4 f ( v - 2 k ) + g ( v - 2 k ) ( f ( v - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ( m - 2 s ) ) - h - j + 2 n ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) h + j ( - ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) 2 f ( v - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 f ( v - 2 k ) - ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) 2 f ( v - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) 2 4 f ( v - 2 k ) ) - b ( m - 2 s ) ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 f ( v - 2 k ) z - b ( m - 2 s ) ) 2 4 f ( v - 2 k ) ) ) + b 2 ( m - 2 s ) 2 4 f ( v - 2 k ) + g ( v - 2 k ) ( f ( v - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ( m - 2 s ) ) - h - j + 2 n ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) h + j ( - ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 s ) ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) + 2 f ( v - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) 2 4 f ( v - 2 k ) ) - ( b ( m - 2 s ) + 2 f ( v - 2 k ) z ) 2 f ( v - 2 k ) ) ) /; n m + v + Condition z z n b z 1 2 m g f z v 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 n 2 -1 m -1 v 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 b m -1 2 k -2 n 1 Binomial m k Gamma 2 n 1 -1 b m -1 2 k z 1 2 Gamma 2 n 1 b m -1 2 k z 1 2 2 -1 m -1 2 n -1 v -1 k 0 v -1 2 -1 Binomial v k s 0 m -1 2 -1 Binomial m s -1 b 2 m -1 2 s 2 4 f v -1 2 k -1 -1 g v -1 2 k h 0 j j 0 n -1 j -1 h 4 j -1 b m -1 2 s -1 h -1 j 2 n -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 h j -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 b m -1 2 s -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 Gamma 1 2 h j 1 -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 -1 2 f v -1 2 k -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 -1 f v -1 2 k -2 n 1 -1 b 2 m -1 2 s 2 4 f v -1 2 k -1 -1 g v -1 2 k h 0 j j 0 n -1 j -1 h 4 j b m -1 2 s -1 h -1 j 2 n b m -1 2 s -1 2 f v -1 2 k z 1 2 h j b m -1 2 s -1 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b m -1 2 s b m -1 2 s -1 2 f v -1 2 k z 1 2 Gamma 1 2 h j 1 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 -1 2 f v -1 2 k b m -1 2 s -1 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 Gamma 1 2 h j 2 b m -1 2 s -1 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 -1 f v -1 2 k -2 n 1 b 2 m -1 2 s 2 4 f v -1 2 k -1 g v -1 2 k f v -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j -1 b m -1 2 s -1 h -1 j 2 n 2 f v -1 2 k z 1 2 -1 b m -1 2 s h j -1 2 f v -1 2 k z 1 2 -1 b m -1 2 s 2 f v -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 f v -1 2 k -1 2 f v -1 2 k z 1 2 -1 b m -1 2 s 2 f v -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 2 f v -1 2 k z 1 2 -1 b m -1 2 s 2 4 f v -1 2 k -1 -1 b m -1 2 s 2 f v -1 2 k z 1 2 -1 b m -1 2 s Gamma 1 2 h j 1 -1 2 f v -1 2 k z 1 2 -1 b m -1 2 s 2 4 f v -1 2 k -1 b 2 m -1 2 s 2 4 f v -1 2 k -1 g v -1 2 k f v -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j b m -1 2 s -1 h -1 j 2 n b m -1 2 s 2 f v -1 2 k z 1 2 h j -1 b m -1 2 s 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b m -1 2 s b m -1 2 s 2 f v -1 2 k z 1 2 Gamma 1 2 h j 1 -1 b m -1 2 s 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 2 f v -1 2 k Gamma 1 2 h j 2 -1 b m -1 2 s 2 f v -1 2 k z 1 2 2 4 f v -1 2 k -1 -1 b m -1 2 s 2 f v -1 2 k z 1 2 2 f v -1 2 k -1 1 2 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_", " ", SqrtBox["z_"]]], "]"]], "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[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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "v"]]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", RowBox[List["2", " ", "n"]], "-", "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["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "-", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "-", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], "+", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], "2"]]], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]], "+", 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["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]]]], "/;", 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