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 Cosh

 http://functions.wolfram.com/01.20.21.4625.01

 Input Form

 Integrate[E^(p z^2) Sinh[b z^2] Cosh[c z^2]^v, z] == 2^(-2 - v) Sqrt[Pi] Binomial[v, v/2] (-(Erfi[Sqrt[-b + p] z]/Sqrt[-b + p]) + Erfi[Sqrt[b + p] z]/Sqrt[b + p]) (1 - Mod[v, 2]) - 2^(-2 - v) Sqrt[Pi] Sum[Binomial[v, s] (Erfi[Sqrt[-b + p + 2 c s - c v] z]/ Sqrt[-b + p + 2 c s - c v] - Erfi[Sqrt[b + p + 2 c s - c v] z]/ Sqrt[b + p + 2 c s - c v] + Erfi[Sqrt[-b + p - 2 c s + c v] z]/ Sqrt[-b + p - 2 c s + c v] - Erfi[Sqrt[b + p - 2 c s + c v] z]/ Sqrt[b + p - 2 c s + c v]), {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[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]]]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["b", "+", "p"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", 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["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]], "-", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["b", "+", "p", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

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

 2002-12-18