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/01.20.21.4652.01

 Input Form

 Integrate[E^(p Sqrt[z]) Sinh[b z]^m Cosh[c z]^v, z] == (1/p^2) (I^m 2^(1 - m - v) E^(p Sqrt[z]) (-1 + p Sqrt[z]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + I^m 2^(-1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k E^((I m Pi)/2) Binomial[m, k] (-((2 E^(p Sqrt[z] - b (-2 k + m) z))/(b (-2 k + m))) + (2 E^((-I) m Pi + p Sqrt[z] + b (-2 k + m) z))/(b (-2 k + m)) - (E^(p^2/(4 b (-2 k + m))) p Sqrt[Pi] Erfi[(p - 2 b (-2 k + m) Sqrt[z])/ (2 Sqrt[(-b) (-2 k + m)])])/((-b) (-2 k + m))^(3/2) - (E^(-(p^2/(4 b (-2 k + m))) - I m Pi) p Sqrt[Pi] Erfi[(p + 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[b (-2 k + m)])])/ (b (-2 k + m))^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}] + I^m 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (-((2 E^(p Sqrt[z] - c (-2 s + v) z))/ (c (-2 s + v))) + (2 E^(p Sqrt[z] + c (-2 s + v) z))/ (c (-2 s + v)) - (E^(p^2/(4 c (-2 s + v))) p Sqrt[Pi] Erfi[(p - 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[(-c) (-2 s + v)])])/ ((-c) (-2 s + v))^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (E^(p^2/(4 c (-2 s + v))) (c (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((2 E^(p Sqrt[z] + ((-b) (2 k - m) - c (-2 s + v)) z))/ ((-b) (2 k - m) - c (-2 s + v)) + (2 E^(I m Pi + p Sqrt[z] + (b (2 k - m) + c (-2 s + v)) z))/ (b (2 k - m) + c (-2 s + v)) - (p Sqrt[Pi] Erfi[(p + 2 ((-b) (2 k - m) - c (-2 s + v)) Sqrt[z])/ (2 Sqrt[(-b) (2 k - m) - c (-2 s + v)])])/ E^(p^2/(4 ((-b) (2 k - m) - c (-2 s + v))))/ ((-b) (2 k - m) - c (-2 s + v))^(3/2) - (E^(I m Pi - p^2/(4 (b (2 k - m) + c (-2 s + v)))) p Sqrt[Pi] Erfi[(p + 2 (b (2 k - m) + c (-2 s + v)) Sqrt[z])/ (2 Sqrt[b (2 k - m) + c (-2 s + v)])])/ (b (2 k - m) + c (-2 s + v))^(3/2) + (-1)^m ((2 E^(p Sqrt[z] + ((-b) (-2 k + m) - c (-2 s + v)) z))/ ((-b) (-2 k + m) - c (-2 s + v)) + (2 E^((-I) m Pi + p Sqrt[z] + (b (-2 k + m) + c (-2 s + v)) z))/ (b (-2 k + m) + c (-2 s + v)) - (p Sqrt[Pi] Erfi[(p + 2 ((-b) (-2 k + m) - c (-2 s + v)) Sqrt[z])/ (2 Sqrt[(-b) (-2 k + m) - c (-2 s + v)])])/ E^(p^2/(4 ((-b) (-2 k + m) - c (-2 s + v))))/ ((-b) (-2 k + m) - c (-2 s + v))^(3/2) - (E^((-I) m Pi - p^2/(4 (b (-2 k + m) + c (-2 s + v)))) p Sqrt[Pi] Erfi[(p + 2 (b (-2 k + m) + c (-2 s + v)) Sqrt[z])/(2 Sqrt[b (-2 k + m) + c (-2 s + v)])])/ (b (-2 k + m) + c (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["p", "2"]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["p", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], "+", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z sinh m ( b z ) cosh v ( c z ) z m 2 - m - v + 1 p z ( p z - 1 ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p 2 + m 2 - m - v - 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k m π 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - p 2 4 b ( m - 2 k ) p π erfi ( p - 2 b ( m - 2 k ) z 2 - b ( m - 2 k ) ) ( - b ( m - 2 k ) ) 3 / 2 + 2 - π m + b ( m - 2 k ) z + p z b ( m - 2 k ) - 2 p z - b ( m - 2 k ) z b ( m - 2 k ) - - p 2 4 b ( m - 2 k ) - m π p π erfi ( 2 b z ( m - 2 k ) + p 2 b ( m - 2 k ) ) ( b ( m - 2 k ) ) 3 / 2 ) + m 2 - m - v - 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 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 c ( v - 2 s ) p π erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 + 2 z p + c ( v - 2 s ) z c ( v - 2 s ) - 2 p z - c ( v - 2 s ) z c ( v - 2 s ) - - p 2 4 c ( v - 2 s ) p π erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 ) + 2 - m - v - 1 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m ( 2 - π m + ( b ( m - 2 k ) + c ( v - 2 s ) ) z + p z b ( m - 2 k ) + c ( v - 2 s ) - - p 2 4 ( b ( m - 2 k ) + c ( v - 2 s ) ) - m π p π erfi ( p + 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) z 2 b ( m - 2 k ) + c ( v - 2 s ) ) ( b ( m - 2 k ) + c ( v - 2 s ) ) 3 / 2 + 2 z p + ( - b ( m - 2 k ) - c ( v - 2 s ) ) z - b ( m - 2 k ) - c ( v - 2 s ) - - p 2 4 ( - b ( m - 2 k ) - c ( v - 2 s ) ) p π erfi ( p + 2 ( - b ( m - 2 k ) - c ( v - 2 s ) ) z 2 - b ( m - 2 k ) - c ( v - 2 s ) ) ( - b ( m - 2 k ) - c ( v - 2 s ) ) 3 / 2 ) + 2 π m + ( b ( 2 k - m ) + c ( v - 2 s ) ) z + p z b ( 2 k - m ) + c ( v - 2 s ) - m π - p 2 4 ( b ( 2 k - m ) + c ( v - 2 s ) ) p π erfi ( p + 2 ( b ( 2 k - m ) + c ( v - 2 s ) ) z 2 b ( 2 k - m ) + c ( v - 2 s ) ) ( b ( 2 k - m ) + c ( v - 2 s ) ) 3 / 2 + 2 z p + ( - b ( 2 k - m ) - c ( v - 2 s ) ) z - b ( 2 k - m ) - c ( v - 2 s ) - - p 2 4 ( - b ( 2 k - m ) - c ( v - 2 s ) ) p π erfi ( p + 2 ( - b ( 2 k - m ) - c ( v - 2 s ) ) z 2 - b ( 2 k - m ) - c ( v - 2 s ) ) ( - b ( 2 k - m ) - c ( v - 2 s ) ) 3 / 2 ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z 1 2 b z m c z v m 2 -1 m -1 v 1 p z 1 2 p z 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p 2 -1 m 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k m 2 -1 Binomial m k -1 p 2 4 b m -1 2 k -1 p 1 2 Erfi p -1 2 b m -1 2 k z 1 2 2 -1 b m -1 2 k 1 2 -1 -1 b m -1 2 k 3 2 -1 2 -1 m b m -1 2 k z p z 1 2 b m -1 2 k -1 -1 2 p z 1 2 -1 b m -1 2 k z b m -1 2 k -1 -1 -1 p 2 4 b m -1 2 k -1 -1 m p 1 2 Erfi 2 b z 1 2 m -1 2 k p 2 b m -1 2 k 1 2 -1 b m -1 2 k 3 2 -1 m 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p -1 2 c v -1 2 s z 1 2 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 2 z 1 2 p c v -1 2 s z c v -1 2 s -1 -1 2 p z 1 2 -1 c v -1 2 s z c v -1 2 s -1 -1 -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 m 2 -1 m b m -1 2 k c v -1 2 s z p z 1 2 b m -1 2 k c v -1 2 s -1 -1 -1 p 2 4 b m -1 2 k c v -1 2 s -1 -1 m p 1 2 Erfi p 2 b m -1 2 k c v -1 2 s z 1 2 2 b m -1 2 k c v -1 2 s 1 2 -1 b m -1 2 k c v -1 2 s 3 2 -1 2 z 1 2 p -1 b m -1 2 k -1 c v -1 2 s z -1 b m -1 2 k -1 c v -1 2 s -1 -1 -1 p 2 4 -1 b m -1 2 k -1 c v -1 2 s -1 p 1 2 Erfi p 2 -1 b m -1 2 k -1 c v -1 2 s z 1 2 2 -1 b m -1 2 k -1 c v -1 2 s 1 2 -1 -1 b m -1 2 k -1 c v -1 2 s 3 2 -1 2 m b 2 k -1 m c v -1 2 s z p z 1 2 b 2 k -1 m c v -1 2 s -1 -1 m -1 p 2 4 b 2 k -1 m c v -1 2 s -1 p 1 2 Erfi p 2 b 2 k -1 m c v -1 2 s z 1 2 2 b 2 k -1 m c v -1 2 s 1 2 -1 b 2 k -1 m c v -1 2 s 3 2 -1 2 z 1 2 p -1 b 2 k -1 m -1 c v -1 2 s z -1 b 2 k -1 m -1 c v -1 2 s -1 -1 -1 p 2 4 -1 b 2 k -1 m -1 c v -1 2 s -1 p 1 2 Erfi p 2 -1 b 2 k -1 m -1 c v -1 2 s z 1 2 2 -1 b 2 k -1 m -1 c v -1 2 s 1 2 -1 -1 b 2 k -1 m -1 c v -1 2 s 3 2 -1 m m 0 v v 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SqrtBox["z_"]]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["p", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], SuperscriptBox["p", "2"]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "+", RowBox[List["p", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "m", " ", "\[Pi]"]], "-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18