html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.1161.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18