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 Cosh

 http://functions.wolfram.com/01.20.21.1302.01

 Input Form

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

 Standard Form

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

 MathML Form

 p z 2 cos m ( b z 2 ) cosh ( c z 2 ) z 2 - m - 2 ( erfi ( p - c z ) p - c + erfi ( c + p z ) c + p ) π ( 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 ) + 2 - m - 2 π k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( erfi ( - c - 2 b k + b m + p z ) - c - 2 b k + b m + p + erfi ( c - 2 b k + b m + p z ) c - 2 b k + b m + p + erfi ( - c + 2 b k - b m + p z ) - c + 2 b k - b m + p + erfi ( c + 2 b k - b m + p z ) c + 2 b k - b m + p ) /; m + Condition z p z 2 b z 2 m c z 2 2 -1 m -2 Erfi p -1 c 1 2 z p -1 c 1 2 -1 Erfi c p 1 2 z c p 1 2 -1 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 2 -1 m -2 1 2 k 0 m -1 2 -1 Binomial m k Erfi -1 c -1 2 b k b m p 1 2 z -1 c -1 2 b k b m p 1 2 -1 Erfi c -1 2 b k b m p 1 2 z c -1 2 b k b m p 1 2 -1 Erfi -1 c 2 b k -1 b m p 1 2 z -1 c 2 b k -1 b m p 1 2 -1 Erfi c 2 b k -1 b m p 1 2 z c 2 b k -1 b m p 1 2 -1 m 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"]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["c", "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["c", "+", "p"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", 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[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]]], "+", FractionBox[RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]], " ", "z"]], "]"]], SqrtBox[RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18