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; }

 Cos

 http://functions.wolfram.com/01.07.21.2550.01

 Input Form

 Integrate[z^n Sin[d z]^m Cos[c Sqrt[z] + f z]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) (2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) + 2 I d^(-1 - 2 n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k (2 k - m)^(-1 - 2 n) Binomial[m, k] ((-E^(I m Pi)) (I d (2 k - m))^n Gamma[1 + n, (-I) d (2 k - m) z] + ((-I) d (2 k - m))^n Gamma[1 + n, I d (2 k - m) z]))/ E^((1/2) I m Pi), {k, 0, Floor[(1/2) (-1 + m)]}] + (1/c) ((I (-(1/4))^n Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[((-2 s + v)^(-2 - 2 n) Binomial[v, s] (Sum[(-1)^(-h + k) 4^k (I c (2 s - v))^(1 - h - k + 2 n) (I (2 s - v) (c + 2 f Sqrt[z]))^(h + k) (-((I (2 s - v) (c + 2 f Sqrt[z])^2)/f))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (2 s - v) (c + 2 f Sqrt[z])^2)/ (4 f))] - 2 I f Sqrt[-((I (2 s - v) (c + 2 f Sqrt[z])^2)/ f)] Gamma[(1/2) (2 + h + k), -((I (2 s - v) (c + 2 f Sqrt[z])^2)/(4 f))]), {k, 0, n}, {h, 0, k}] - E^((I c^2 (2 s - v))/(2 f)) Sum[(-1)^(-h + k) 4^k ((-I) c (2 s - v))^(1 - h - k + 2 n) ((-I) (2 s - v) (c + 2 f Sqrt[z]))^(h + k) ((I (2 s - v) (c + 2 f Sqrt[z])^2)/ f)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (2 s - v) (c + 2 f Sqrt[z])^2)/(4 f)] + 2 I f Sqrt[(I (2 s - v) (c + 2 f Sqrt[z])^2)/f] Gamma[ (1/2) (2 + h + k), (I (2 s - v) (c + 2 f Sqrt[z])^2)/ (4 f)]), {k, 0, n}, {h, 0, k}]))/ E^((I c^2 (2 s - v))/(4 f)), {s, 0, Floor[(1/2) (-1 + v)]}])/ f^(2 (1 + n))) - Sum[(-1)^u Binomial[m, u] Sum[Binomial[v, s] (Sum[(-1)^(-h + k) 4^k (I (2 c s - c v))^(-h - k + 2 n) (I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z]))^(h + k) (-((I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v)))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-2 c s + c v) (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v))] + 2 I (d m + 2 f s - 2 d u - f v) Sqrt[-((I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v))] Gamma[(1/2) (2 + h + k), -((I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v))]), {k, 0, n}, {h, 0, k}]/ (E^((I (2 c s - c v)^2)/(4 d m + 8 f s - 8 d u - 4 f v)) (I (d m + 2 f s - 2 d u - f v))^(2 n))/ (d m + 2 f s - 2 d u - f v)^2 + ((-1)^m E^((I (2 c s - c v)^2)/(4 d m + 8 f s - 8 d u - 4 f v)) Sum[(-1)^(-h + k) 4^k ((-I) (2 c s - c v))^(-h - k + 2 n) ((-I) (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z]))^ (h + k) ((I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-2 c s + c v) (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v)] - 2 I (d m + 2 f s - 2 d u - f v) Sqrt[(I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(d m + 2 f s - 2 d u - f v)] Gamma[(1/2) (2 + h + k), (I (c (2 s - v) + 2 (d m + 2 f s - 2 d u - f v) Sqrt[z])^2)/(4 d m + 8 f s - 8 d u - 4 f v)]), {k, 0, n}, {h, 0, k}])/ ((-I) (d m + 2 f s - 2 d u - f v))^(2 n)/ (d m + 2 f s - 2 d u - f v)^2 + (((d (m - 2 u) + f (-2 s + v))^2)^ (-1 - 2 n) (((-I) (d (m - 2 u) + f (-2 s + v)))^(2 n) Sum[(-1)^(-h + k) 4^k (I c (-2 s + v))^(-h - k + 2 n) (I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]))^ (h + k) (-((I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + f (-2 s + v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-c) (-2 s + v) (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v))))] + 2 I (d (m - 2 u) + f (-2 s + v)) Sqrt[-((I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/ (d (m - 2 u) + f (-2 s + v)))] Gamma[(1/2) (2 + h + k), -((I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(4 (d (m - 2 u) + f (-2 s + v))))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I c^2 (-2 s + v)^2)/ (2 (d (m - 2 u) + f (-2 s + v)))) (I (d (m - 2 u) + f (-2 s + v)))^(2 n) Sum[(-1)^(-h + k) 4^k ((-I) c (-2 s + v))^(-h - k + 2 n) ((-I) (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]))^(h + k) ((I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^ 2)/(d (m - 2 u) + f (-2 s + v)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-c) (-2 s + v) (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/ (4 (d (m - 2 u) + f (-2 s + v)))] - 2 I (d (m - 2 u) + f (-2 s + v)) Sqrt[(I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^2)/(d (m - 2 u) + f (-2 s + v))] Gamma[(1/2) (2 + h + k), (I (c (-2 s + v) + 2 (d (m - 2 u) + f (-2 s + v)) Sqrt[z])^ 2)/(4 (d (m - 2 u) + f (-2 s + v)))]), {k, 0, n}, {h, 0, k}]))/E^((I c^2 (-2 s + v)^2)/(4 (d (m - 2 u) + f (-2 s + v))))), {s, 0, Floor[(1/2) (-1 + v)]}], {u, 0, Floor[(1/2) (-1 + m)]}]/(I^m 4^n)) /; 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["Sin", "[", RowBox[List["d", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "+", "n"]]], RowBox[List["(", RowBox[List["2", " ", 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[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["1", "c"], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "4"]]], ")"]], "n"], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], RowBox[List["4", " ", "f"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List["1", "-", "h", "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], RowBox[List["2", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List["1", "-", "h", "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "u"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "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", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "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 sin m ( d z ) cos v ( z c + f z ) z 2 - m - v - 1 ( 2 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 2 d - 2 n - 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 ( - 1 ) k - 1 2 m π ( 2 k - m ) - 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]]]]] ( ( - d ( 2 k - m ) ) n Γ ( n + 1 , d ( 2 k - m ) z ) - m π ( d ( 2 k - m ) ) n Γ ( n + 1 , - d ( 2 k - m ) z ) ) + 1 c ( ( - 1 4 ) n f - 2 ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 - c 2 ( 2 s - v ) 4 f ( v - 2 s ) - 2 n - 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n + 1 ( ( 2 s - v ) ( c + 2 f z ) ) h + k ( - ( 2 s - v ) ( c + 2 f z ) 2 f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( 2 s - v ) ( c + 2 f z ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 s - v ) ( c + 2 f z ) 2 4 f ) - 2 f - ( 2 s - v ) ( c + 2 f z ) 2 f Γ ( 1 2 ( h + k + 2 ) , - ( 2 s - v ) ( c + 2 f z ) 2 4 f ) ) - c 2 ( 2 s - v ) 2 f k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - c ( 2 s - v ) ) - h - k + 2 n + 1 ( - ( 2 s - v ) ( c + 2 f z ) ) h + k ( ( 2 s - v ) ( c + 2 f z ) 2 f ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( 2 s - v ) ( c + 2 f z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 s - v ) ( c + 2 f z ) 2 4 f ) + 2 ( 2 s - v ) ( c + 2 f z ) 2 f f Γ ( 1 2 ( h + k + 2 ) , ( 2 s - v ) ( c + 2 f z ) 2 4 f ) ) ) ) - - m 4 - n u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( - 1 ) m ( 2 c s - c v ) 2 4 d m + 8 f s - 8 d u - 4 f v ( - ( d m + 2 f s - 2 d u - f v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( 2 c s - c v ) ) - h - k + 2 n ( - ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) ) h + k ( ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 d m + 2 f s - 2 d u - f v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c v - 2 c s ) ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 4 d m + 8 f s - 8 d u - 4 f v ) - 2 ( d m + 2 f s - 2 d u - f v ) ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 d m + 2 f s - 2 d u - f v Γ ( 1 2 ( h + k + 2 ) , ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 4 d m + 8 f s - 8 d u - 4 f v ) ) ) / ( d m + 2 f s - 2 d u - f v ) 2 + ( - ( 2 c s - c v ) 2 4 d m + 8 f s - 8 d u - 4 f v ( ( d m + 2 f s - 2 d u - f v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( ( 2 c s - c v ) ) - h - k + 2 n ( ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) ) h + k ( - ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 d m + 2 f s - 2 d u - f v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c v - 2 c s ) ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 4 d m + 8 f s - 8 d u - 4 f v ) + 2 ( d m + 2 f s - 2 d u - f v ) Γ ( 1 2 ( h + k + 2 ) , - ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 4 d m + 8 f s - 8 d u - 4 f v ) ( - ( ( c ( 2 s - v ) + 2 ( d m + 2 f s - 2 d u - f v ) z ) 2 ) / ( d m + 2 f s - 2 d u - f v ) ) ) ) / ( d m + 2 f s - 2 d u - f v ) 2 + - c 2 ( v - 2 s ) 2 4 ( d ( m - 2 u ) + f ( v - 2 s ) ) ( ( d ( m - 2 u ) + f ( v - 2 s ) ) 2 ) - 2 n - 1 ( ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) ) h + k ( - ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 ) / ( d ( m - 2 u ) + f ( v - 2 s ) ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) ( - ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 ) / ( d ( m - 2 u ) + f ( v - 2 s ) ) ) Γ ( 1 2 ( h + k + 2 ) , - ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 ) / ( 4 ( d ( m - 2 u ) + f ( v - 2 s ) ) ) ) - c ( v - 2 s ) ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 ) / ( 4 ( d ( m - 2 u ) + f ( v - 2 s ) ) ) ) ) ) ( - ( d ( m - 2 u ) + f ( v - 2 s ) ) ) 2 n + ( - 1 ) m c 2 ( v - 2 s ) 2 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) ( ( d ( m - 2 u ) + f ( v - 2 s ) ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - c ( v - 2 s ) ) - h - k + 2 n ( - ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) ) h + k ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 d ( m - 2 u ) + f ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - c ( v - 2 s ) ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 4 ( d ( m - 2 u ) + f ( v - 2 s ) ) ) - 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) ( ( ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 ) / ( d ( m - 2 u ) + f ( v - 2 s ) ) ) Γ ( 1 2 ( h + k + 2 ) , ( c ( v - 2 s ) + 2 ( d ( m - 2 u ) + f ( v - 2 s ) ) z ) 2 4 ( d ( m - 2 u ) + f ( v - 2 s ) ) ) ) ) ) ) /; n m + v + Condition z z n d z m z 1 2 c f z v 2 -1 m -1 v -1 2 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 2 d -2 n -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k -1 1 2 m 2 k -1 m -2 n -1 Binomial m k -1 d 2 k -1 m n Gamma n 1 d 2 k -1 m z -1 m d 2 k -1 m n Gamma n 1 -1 d 2 k -1 m z 1 c -1 -1 1 4 n f -2 n 1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 -1 c 2 2 s -1 v 4 f -1 v -1 2 s -2 n -2 Binomial v s h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n 1 2 s -1 v c 2 f z 1 2 h k -1 2 s -1 v c 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v c 2 f z 1 2 Gamma 1 2 h k 1 -1 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 2 f -1 2 s -1 v c 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h k 2 -1 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 c 2 2 s -1 v 2 f -1 h 0 k k 0 n -1 k -1 h 4 k -1 c 2 s -1 v -1 h -1 k 2 n 1 -1 2 s -1 v c 2 f z 1 2 h k 2 s -1 v c 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v c 2 f z 1 2 Gamma 1 2 h k 1 2 s -1 v c 2 f z 1 2 2 4 f -1 2 2 s -1 v c 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h k 2 2 s -1 v c 2 f z 1 2 2 4 f -1 -1 -1 m 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 Binomial v s -1 m 2 c s -1 c v 2 4 d m 8 f s -1 8 d u -1 4 f v -1 -1 d m 2 f s -1 2 d u -1 f v -2 n h 0 k k 0 n -1 k -1 h 4 k -1 2 c s -1 c v -1 h -1 k 2 n -1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 h k c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 d m 2 f s -1 2 d u -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 Gamma 1 2 h k 1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 4 d m 8 f s -1 8 d u -1 4 f v -1 -1 2 d m 2 f s -1 2 d u -1 f v c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 d m 2 f s -1 2 d u -1 f v -1 1 2 Gamma 1 2 h k 2 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 4 d m 8 f s -1 8 d u -1 4 f v -1 d m 2 f s -1 2 d u -1 f v 2 -1 -1 2 c s -1 c v 2 4 d m 8 f s -1 8 d u -1 4 f v -1 d m 2 f s -1 2 d u -1 f v -2 n h 0 k k 0 n -1 k -1 h 4 k 2 c s -1 c v -1 h -1 k 2 n c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 h k -1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 d m 2 f s -1 2 d u -1 f v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 Gamma 1 2 h k 1 -1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 4 d m 8 f s -1 8 d u -1 4 f v -1 2 d m 2 f s -1 2 d u -1 f v Gamma 1 2 h k 2 -1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 4 d m 8 f s -1 8 d u -1 4 f v -1 -1 c 2 s -1 v 2 d m 2 f s -1 2 d u -1 f v z 1 2 2 d m 2 f s -1 2 d u -1 f v -1 d m 2 f s -1 2 d u -1 f v 2 -1 -1 c 2 v -1 2 s 2 4 d m -1 2 u f v -1 2 s -1 d m -1 2 u f v -1 2 s 2 -2 n -1 h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 h k -1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 d m -1 2 u f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d m -1 2 u f v -1 2 s -1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 d m -1 2 u f v -1 2 s -1 Gamma 1 2 h k 2 -1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 4 d m -1 2 u f v -1 2 s -1 -1 c v -1 2 s c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 4 d m -1 2 u f v -1 2 s -1 -1 d m -1 2 u f v -1 2 s 2 n -1 m c 2 v -1 2 s 2 2 d m -1 2 u f v -1 2 s -1 d m -1 2 u f v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k -1 c v -1 2 s -1 h -1 k 2 n -1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 h k c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 d m -1 2 u f v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 c v -1 2 s c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 Gamma 1 2 h k 1 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 4 d m -1 2 u f v -1 2 s -1 -1 2 d m -1 2 u f v -1 2 s c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 d m -1 2 u f v -1 2 s -1 Gamma 1 2 h k 2 c v -1 2 s 2 d m -1 2 u f v -1 2 s z 1 2 2 4 d m -1 2 u f v -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["Sin", "[", RowBox[List["d_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", 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[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["1", "+", "n"]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "4"]]], ")"]], "n"], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], RowBox[List["4", " ", "f"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List["1", "-", "h", "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], RowBox[List["2", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List["1", "-", "h", "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "f"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]], "c"], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "u"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "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", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["4", " ", "d", " ", "m"]], "+", RowBox[List["8", " ", "f", " ", "s"]], "-", RowBox[List["8", " ", "d", " ", "u"]], "-", RowBox[List["4", " ", "f", " ", "v"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "f", " ", "s"]], "-", RowBox[List["2", " ", "d", " ", "u"]], "-", RowBox[List["f", " ", "v"]]]], ")"]], "2"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "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