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/01.20.21.3915.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z] Cosh[c z]^v, z] == (-I) 2^(-2 - 2 n - v) (I b)^(-2 - 2 n) E^((I p^2)/(4 b)) Binomial[v, v/2] (1 - Mod[v, 2]) ((-E^(-((I p^2)/(2 b)))) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 I b Sqrt[z])^(h + k) (-((I (p - 2 I b Sqrt[z])^2)/b))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (p - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (p - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + k), -((I (p - 2 I b Sqrt[z])^2)/(4 b))]), {k, 0, n}, {h, 0, k}] + Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 I b Sqrt[z])^(h + k) ((I (p + 2 I b Sqrt[z])^2)/b)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (p + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (p + 2 I b Sqrt[z])^2)/(4 b)]), {k, 0, n}, {h, 0, k}]) - I 2^(-2 - 2 n - v) Sum[Binomial[v, s] ((E^(p^2/(-4 I b - 8 c s + 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 (I b + 2 c s - c v) Sqrt[z])^(h + k) (-((p + 2 (I b + 2 c s - c v) Sqrt[z])^2/(I b + 2 c s - c v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 (I b + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((p + 2 (I b + 2 c s - c v) Sqrt[z])^2/(4 I b + 8 c s - 4 c v))] + 2 (I b + 2 c s - c v) Sqrt[-((p + 2 (I b + 2 c s - c v) Sqrt[z])^2/(I b + 2 c s - c v))] Gamma[(1/2) (2 + h + k), -((p + 2 (I b + 2 c s - c v) Sqrt[z])^2/(4 I b + 8 c s - 4 c v))]), {k, 0, n}, {h, 0, k}])/(I b + 2 c s - c v)^(2 (1 + n)) - (E^(p^2/(4 I b + 8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 ((-I) b - 2 c s + c v) Sqrt[z])^(h + k) ((p + 2 ((-I) b - 2 c s + c v) Sqrt[z])^2/(I b + 2 c s - c v))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 ((-I) b - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p + 2 ((-I) b - 2 c s + c v) Sqrt[z])^2/(4 I b + 8 c s - 4 c v)] + 2 ((-I) b - 2 c s + c v) Sqrt[(p + 2 ((-I) b - 2 c s + c v) Sqrt[z])^2/(I b + 2 c s - c v)] Gamma[(1/2) (2 + h + k), (p + 2 ((-I) b - 2 c s + c v) Sqrt[z])^2/(4 I b + 8 c s - 4 c v)]), {k, 0, n}, {h, 0, k}])/((-I) b - 2 c s + c v)^(2 (1 + n)) - (E^(p^2/(4 I b - 8 c s + 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 (I b - 2 c s + c v) Sqrt[z])^(h + k) ((p - 2 (I b - 2 c s + c v) Sqrt[z])^2/(I b - 2 c s + c v))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 (I b - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p - 2 (I b - 2 c s + c v) Sqrt[z])^2/(4 (I b - 2 c s + c v))] - 2 (I b - 2 c s + c v) Sqrt[(p - 2 (I b - 2 c s + c v) Sqrt[z])^2/(I b - 2 c s + c v)] Gamma[(1/2) (2 + h + k), (p - 2 (I b - 2 c s + c v) Sqrt[z])^2/( 4 (I b - 2 c s + c v))]), {k, 0, n}, {h, 0, k}])/ ((-I) b + 2 c s - c v)^(2 (1 + n)) + (E^(p^2/(-4 I b + 8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 (I b - 2 c s + c v) Sqrt[z])^(h + k) (-((p + 2 (I b - 2 c s + c v) Sqrt[z])^2/(I b - 2 c s + c v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 (I b - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((p + 2 (I b - 2 c s + c v) Sqrt[z])^2/(4 (I b - 2 c s + c v)))] + 2 (I b - 2 c s + c v) Sqrt[-((p + 2 (I b - 2 c s + c v) Sqrt[z])^2/(I b - 2 c s + c v))] Gamma[(1/2) (2 + h + k), -((p + 2 (I b - 2 c s + c v) Sqrt[z])^2/(4 (I b - 2 c s + c v)))]), {k, 0, n}, {h, 0, k}])/(I b - 2 c s + c v)^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["4", " ", "b"]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["2", " ", "b"]]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z sin ( b z ) cosh v ( c z ) z - 2 - 2 n - v - 2 p 2 4 b ( 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 ) ( b ) - 2 n - 2 ( k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 b z ) h + k ( ( p + 2 b z ) 2 b ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 b z ) 2 4 b ) + 2 ( p + 2 b z ) 2 b b Γ ( 1 2 ( h + k + 2 ) , ( p + 2 b z ) 2 4 b ) ) - - p 2 2 b k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 b z ) h + k ( - ( p - 2 b z ) 2 b ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( p - 2 b z ) 2 4 b ) - 2 b - ( p - 2 b z ) 2 b Γ ( 1 2 ( h + k + 2 ) , - ( p - 2 b z ) 2 4 b ) ) ) - 2 - 2 n - v - 2 s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - p 2 4 b + 8 c s - 4 c v ( - b - 2 c s + c v ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( - b - 2 c s + c v ) z ) h + k ( ( p + 2 ( - b - 2 c s + c v ) z ) 2 b + 2 c s - c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 ( - b - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( - b - 2 c s + c v ) z ) 2 4 b + 8 c s - 4 c v ) + 2 ( - b - 2 c s + c v ) ( p + 2 ( - b - 2 c s + c v ) z ) 2 b + 2 c s - c v Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( - b - 2 c s + c v ) z ) 2 4 b + 8 c s - 4 c v ) ) - p 2 4 b - 8 c s + 4 c v ( - b + 2 c s - c v ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 ( b - 2 c s + c v ) z ) h + k ( ( p - 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 ( b - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p - 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) - 2 ( b - 2 c s + c v ) ( p - 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v Γ ( 1 2 ( h + k + 2 ) , ( p - 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) ) + p 2 - 4 b + 8 c s - 4 c v ( b - 2 c s + c v ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( b - 2 c s + c v ) z ) h + k ( - ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 ( b - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) + 2 - ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ( b - 2 c s + c v ) Γ ( 1 2 ( h + k + 2 ) , - ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) ) + p 2 - 4 b - 8 c s + 4 c v ( b + 2 c s - c v ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( b + 2 c s - c v ) z ) h + k ( - ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 ( b + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p + 2 ( b + 2 c s - c v ) z ) 2 4 b + 8 c s - 4 c v ) + 2 - ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ( b + 2 c s - c v ) Γ ( 1 2 ( h + k + 2 ) , - ( p + 2 ( b + 2 c s - c v ) z ) 2 4 b + 8 c s - 4 c v ) ) ) /; v + n Condition z z n p z 1 2 b z c z v -1 2 -2 n -1 v -2 p 2 4 b -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b -2 n -2 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b z 1 2 h k p 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b z 1 2 Gamma 1 2 h k 1 p 2 b z 1 2 2 4 b -1 2 p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h k 2 p 2 b z 1 2 2 4 b -1 -1 -1 p 2 2 b -1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 b z 1 2 h k -1 p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 b z 1 2 Gamma 1 2 h k 1 -1 p -1 2 b z 1 2 2 4 b -1 -1 2 b -1 p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h k 2 -1 p -1 2 b z 1 2 2 4 b -1 -1 2 -2 n -1 v -2 s 0 v -1 2 -1 Binomial v s -1 p 2 4 b 8 c s -1 4 c v -1 -1 b -1 2 c s c v -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 -1 b -1 2 c s c v z 1 2 h k p 2 -1 b -1 2 c s c v z 1 2 2 b 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 -1 b -1 2 c s c v z 1 2 Gamma 1 2 h k 1 p 2 -1 b -1 2 c s c v z 1 2 2 4 b 8 c s -1 4 c v -1 2 -1 b -1 2 c s c v p 2 -1 b -1 2 c s c v z 1 2 2 b 2 c s -1 c v -1 1 2 Gamma 1 2 h k 2 p 2 -1 b -1 2 c s c v z 1 2 2 4 b 8 c s -1 4 c v -1 -1 p 2 4 b -1 8 c s 4 c v -1 -1 b 2 c s -1 c v -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 b -1 2 c s c v z 1 2 h k p -1 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 b -1 2 c s c v z 1 2 Gamma 1 2 h k 1 p -1 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 -1 2 b -1 2 c s c v p -1 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 Gamma 1 2 h k 2 p -1 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 p 2 -4 b 8 c s -1 4 c v -1 b -1 2 c s c v -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b -1 2 c s c v z 1 2 h k -1 p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b -1 2 c s c v z 1 2 Gamma 1 2 h k 1 -1 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 2 -1 p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 b -1 2 c s c v Gamma 1 2 h k 2 -1 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 p 2 -4 b -1 8 c s 4 c v -1 b 2 c s -1 c v -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b 2 c s -1 c v z 1 2 h k -1 p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 -1 p 2 b 2 c s -1 c v z 1 2 2 4 b 8 c s -1 4 c v -1 2 -1 p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 b 2 c s -1 c v Gamma 1 2 h k 2 -1 p 2 b 2 c s -1 c v z 1 2 2 4 b 8 c s -1 4 c v -1 v SuperPlus n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SqrtBox["z_"]]]], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["4", " ", "b"]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["2", " ", "b"]]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]], "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["p", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18