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/01.20.21.4502.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18