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 Cosh

 http://functions.wolfram.com/01.20.21.0654.01

 Input Form

 Integrate[Sin[d z]^m Cosh[c z^2 + f z], z] == (1/Sqrt[I c]) (2^(-(1/2) - m) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) (I Cosh[f^2/(4 c)] FresnelC[(f + 2 c z)/(Sqrt[I c] Sqrt[2 Pi])] + FresnelS[(f + 2 c z)/(Sqrt[I c] Sqrt[2 Pi])] Sinh[f^2/(4 c)])) + 2^(-(1/2) - m) Sqrt[Pi] Sum[(-1)^(k + m) Binomial[m, k] ((1/Sqrt[(-I) c]) (Cos[(I ((-I) f + 2 d k - d m)^2)/(4 c) + (m Pi)/2] FresnelC[((-I) f + 2 d k - d m - 2 I c z)/(Sqrt[(-I) c] Sqrt[2 Pi])] + FresnelS[((-I) f + 2 d k - d m - 2 I c z)/ (Sqrt[(-I) c] Sqrt[2 Pi])] Sin[(I ((-I) f + 2 d k - d m)^2)/ (4 c) + (m Pi)/2]) + (1/Sqrt[I c]) (Cosh[(I f + 2 d k - d m)^2/(4 c) + (I m Pi)/2] FresnelC[(I f + 2 d k - d m + 2 I c z)/(Sqrt[I c] Sqrt[2 Pi])] - I FresnelS[(I f + 2 d k - d m + 2 I c z)/(Sqrt[I c] Sqrt[2 Pi])] Sinh[(I f + 2 d k - d m)^2/(4 c) + (I m Pi)/2])), {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["d", " ", "z"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]], 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", "[", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", "c"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", "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", 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SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"]]], RowBox[List["4", " ", "c"]]], "+", FractionBox[RowBox[List["m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sin m ( d z ) cosh ( c z 2 + f z ) z 2 - m - 1 2 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 ) ( cosh ( f 2 4 c ) C TagBox["C", FresnelC] ( f + 2 c z c 2 π ) + S TagBox["S", FresnelS] ( f + 2 c z c 2 π ) sinh ( f 2 4 c ) ) c + 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 ( cos ( ( - f + 2 d k - d m ) 2 4 c + m π 2 ) C TagBox["C", FresnelC] ( - f + 2 d k - d m - 2 c z - c 2 π ) + S TagBox["S", FresnelS] ( - f + 2 d k - d m - 2 c z - c 2 π ) sin ( ( - f + 2 d k - d m ) 2 4 c + m π 2 ) ) + 1 c ( cosh ( ( f + 2 d k - d m ) 2 4 c + m π 2 ) C TagBox["C", FresnelC] ( f + 2 d k - d m + 2 c z c 2 π ) - S TagBox["S", FresnelS] ( f + 2 d k - d m + 2 c z c 2 π ) sinh ( ( f + 2 d k - d m ) 2 4 c + m π 2 ) ) ) /; m + Condition z d z m c z 2 f z 2 -1 m -1 1 2 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 f 2 4 c -1 FresnelC f 2 c z c 1 2 2 1 2 -1 FresnelS f 2 c z c 1 2 2 1 2 -1 f 2 4 c -1 c 1 2 -1 2 -1 m -1 1 2 1 2 k 0 m -1 2 -1 -1 k m Binomial m k 1 -1 c 1 2 -1 -1 f 2 d k -1 d m 2 4 c -1 m 2 -1 FresnelC -1 f 2 d k -1 d m -1 2 c z -1 c 1 2 2 1 2 -1 FresnelS -1 f 2 d k -1 d m -1 2 c z -1 c 1 2 2 1 2 -1 -1 f 2 d k -1 d m 2 4 c -1 m 2 -1 1 c 1 2 -1 f 2 d k -1 d m 2 4 c -1 m 2 -1 FresnelC f 2 d k -1 d m 2 c z c 1 2 2 1 2 -1 -1 FresnelS f 2 d k -1 d m 2 c z c 1 2 2 1 2 -1 f 2 d k -1 d m 2 4 c -1 m 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18