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 Cosh

 http://functions.wolfram.com/01.20.21.0645.01

 Input Form

 Integrate[Sin[b z^2]^m Cosh[c z^2 + g], z] == (1/Sqrt[I c]) (2^(-(1/2) - m) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) (I Cosh[g] FresnelC[(c Sqrt[2/Pi] z)/Sqrt[I c]] - FresnelS[(c Sqrt[2/Pi] z)/Sqrt[I c]] Sinh[g])) + 2^(-(1/2) - m) Sqrt[Pi] Sum[(-1)^(k + m) Binomial[m, k] ((Cos[I g + (m Pi)/2] FresnelC[Sqrt[(-I) c + 2 b k - b m] Sqrt[2/Pi] z] + FresnelS[Sqrt[(-I) c + 2 b k - b m] Sqrt[2/Pi] z] Sin[I g + (m Pi)/2])/Sqrt[(-I) c + 2 b k - b m] + (Cosh[g + (I m Pi)/2] FresnelC[Sqrt[I c + 2 b k - b m] Sqrt[2/Pi] z] - I FresnelS[Sqrt[I c + 2 b k - b m] Sqrt[2/Pi] z] Sinh[g + (I m Pi)/2])/Sqrt[I c + 2 b k - b m]), {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[RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", "g", "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["c", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List["c", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "]"]], " ", RowBox[List["Sinh", "[", "g", "]"]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["m", " ", "\[Pi]"]], "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sin m ( b z 2 ) cosh ( c z 2 + g ) z 2 - m - 1 2 π c ( 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 ) ( cosh ( g ) C TagBox["C", FresnelC] ( c 2 π z c ) - S TagBox["S", FresnelS] ( c 2 π z c ) sinh ( g ) ) + 2 - m - 1 2 π k = 0 m - 1 2 ( - 1 ) k + m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - c + 2 b k - b m ( cos ( g + m π 2 ) C TagBox["C", FresnelC] ( - c + 2 b k - b m 2 π z ) + S TagBox["S", FresnelS] ( - c + 2 b k - b m 2 π z ) sin ( g + m π 2 ) ) + 1 c + 2 b k - b m ( cosh ( g + m π 2 ) C TagBox["C", FresnelC] ( c + 2 b k - b m 2 π z ) - S TagBox["S", FresnelS] ( c + 2 b k - b m 2 π z ) sinh ( g + m π 2 ) ) ) /; m + Condition z b z 2 m c z 2 g 2 -1 m -1 1 2 1 2 c 1 2 -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 g FresnelC c 2 -1 1 2 z c 1 2 -1 -1 FresnelS c 2 -1 1 2 z c 1 2 -1 g 2 -1 m -1 1 2 1 2 k 0 m -1 2 -1 -1 k m Binomial m k 1 -1 c 2 b k -1 b m 1 2 -1 g m 2 -1 FresnelC -1 c 2 b k -1 b m 1 2 2 -1 1 2 z FresnelS -1 c 2 b k -1 b m 1 2 2 -1 1 2 z g m 2 -1 1 c 2 b k -1 b m 1 2 -1 g m 2 -1 FresnelC c 2 b k -1 b m 1 2 2 -1 1 2 z -1 FresnelS c 2 b k -1 b m 1 2 2 -1 1 2 z g m 2 -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", "g", "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["c", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List["c", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "]"]], " ", RowBox[List["Sinh", "[", "g", "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["m", " ", "\[Pi]"]], "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18