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/01.07.21.2537.01

 Input Form

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