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 Cosh

 http://functions.wolfram.com/01.20.21.4062.01

 Input Form

 Integrate[Sinh[b z] Cosh[c Sqrt[z]]^v, z] == (Cosh[b z] Binomial[v, v/2] (1 - Mod[v, 2]))/(2^v b) + 2^(-2 - v) Sum[Binomial[v, s] ((8 Cosh[c (-2 s + v) Sqrt[z]] Cosh[b z])/ b + (c E^((c^2 (-2 s + v)^2)/(4 b)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) - 2 b Sqrt[z])/(2 Sqrt[-b])])/(-b)^(3/2) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) - 2 b Sqrt[z])/(2 Sqrt[b])])/ (E^((c^2 (-2 s + v)^2)/(4 b)) b^(3/2)) + (c E^((c^2 (-2 s + v)^2)/(4 b)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b Sqrt[z])/(2 Sqrt[-b])])/(-b)^(3/2) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b Sqrt[z])/(2 Sqrt[b])])/ (E^((c^2 (-2 s + v)^2)/(4 b)) b^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "b"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "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[FractionBox[RowBox[List["8", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "b"], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z ) cosh v ( c z ) z 2 - v ( 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]]]]] cosh ( b z ) ( 1 - v mod 2 \$CellContext`v 2 ) b + 2 - 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]]]]] ( 8 cosh ( c ( v - 2 s ) z ) cosh ( b z ) b + c c 2 ( v - 2 s ) 2 4 b π ( v - 2 s ) erfi ( 2 z b + c ( v - 2 s ) 2 - b ) ( - b ) 3 / 2 + c c 2 ( v - 2 s ) 2 4 b π ( v - 2 s ) erfi ( c ( v - 2 s ) - 2 b z 2 - b ) ( - b ) 3 / 2 - c - c 2 ( v - 2 s ) 2 4 b π ( v - 2 s ) erfi ( 2 z b + c ( v - 2 s ) 2 b ) b 3 / 2 - c - c 2 ( v - 2 s ) 2 4 b π ( v - 2 s ) erfi ( c ( v - 2 s ) - 2 b z 2 b ) b 3 / 2 ) /; v + Condition z b z c z 1 2 v 2 -1 v Binomial v v 2 -1 b z 1 -1 \$CellContext`v 2 b -1 2 -1 v -2 s 0 v -1 2 -1 Binomial v s 8 c v -1 2 s z 1 2 b z b -1 c c 2 v -1 2 s 2 4 b -1 1 2 v -1 2 s Erfi 2 z 1 2 b c v -1 2 s 2 -1 b 1 2 -1 -1 b 3 2 -1 c c 2 v -1 2 s 2 4 b -1 1 2 v -1 2 s Erfi c v -1 2 s -1 2 b z 1 2 2 -1 b 1 2 -1 -1 b 3 2 -1 -1 c -1 c 2 v -1 2 s 2 4 b -1 1 2 v -1 2 s Erfi 2 z 1 2 b c v -1 2 s 2 b 1 2 -1 b 3 2 -1 -1 c -1 c 2 v -1 2 s 2 4 b -1 1 2 v -1 2 s Erfi c v -1 2 s -1 2 b z 1 2 2 b 1 2 -1 b 3 2 -1 v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "b"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "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[FractionBox[RowBox[List["8", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "b"], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "b"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18