html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.0809.01

 Input Form

 Integrate[Cos[d z]^m Cosh[c Sqrt[z] + g], z] == -((1/c^2) (2^(1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) (Cosh[g + c Sqrt[z]] - c Sqrt[z] Sinh[g + c Sqrt[z]]))) + 2^(-1 - m) Sum[Binomial[m, k] ((1/(2 d k - d m)^(3/2)) ((-I) c Sqrt[2 Pi] Cosh[g - (I c^2)/(4 (2 d k - d m))] FresnelC[(I c + 2 (2 d k - d m) Sqrt[z])/(Sqrt[2 d k - d m] Sqrt[2 Pi])] - c Sqrt[2 Pi] FresnelS[ (I c + 2 (2 d k - d m) Sqrt[z])/(Sqrt[2 d k - d m] Sqrt[2 Pi])] Sinh[g - (I c^2)/(4 (2 d k - d m))] + 2 I Sqrt[2 d k - d m] Sinh[g + c Sqrt[z] - I (2 d k - d m) z]) + (1/(2 d k - d m)^(3/2)) (I c Sqrt[2 Pi] Cos[(-I) g + c^2/(4 (2 d k - d m))] FresnelC[((-I) c + 2 (2 d k - d m) Sqrt[z])/(Sqrt[2 d k - d m] Sqrt[2 Pi])] - I c Sqrt[2 Pi] FresnelS[ ((-I) c + 2 (2 d k - d m) Sqrt[z])/(Sqrt[2 d k - d m] Sqrt[2 Pi])] Sin[(-I) g + c^2/(4 (2 d k - d m))] - 2 I Sqrt[2 d k - d m] Sinh[g + c Sqrt[z] + I (2 d k - d m) z])), {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["Cos", "[", RowBox[List["d", " ", "z"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox["c", "2"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", 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["Cosh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["c", " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", 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["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cosh", "[", RowBox[List["g", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["c", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", 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FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "g"]], "+", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( d z ) cosh ( z c + g ) z 2 - m - 1 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]]]]] ( 1 ( 2 d k - d m ) 3 / 2 ( c 2 π cos ( c 2 4 ( 2 d k - d m ) - g ) C TagBox["C", FresnelC] ( - c + 2 ( 2 d k - d m ) z 2 d k - d m 2 π ) - c 2 π S TagBox["S", FresnelS] ( - c + 2 ( 2 d k - d m ) z 2 d k - d m 2 π ) sin ( c 2 4 ( 2 d k - d m ) - g ) - 2 2 d k - d m sinh ( z c + g + ( 2 d k - d m ) z ) ) + 1 ( 2 d k - d m ) 3 / 2 ( - c 2 π cosh ( - c 2 4 ( 2 d k - d m ) + g ) C TagBox["C", FresnelC] ( c + 2 ( 2 d k - d m ) z 2 d k - d m 2 π ) - c 2 π S TagBox["S", FresnelS] ( c + 2 ( 2 d k - d m ) z 2 d k - d m 2 π ) sinh ( - c 2 4 ( 2 d k - d m ) + g ) + 2 2 d k - d m sinh ( z c + g - ( 2 d k - d m ) z ) ) ) - 2 1 - m ( 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 ( z c + g ) - c z sinh ( z c + g ) ) c 2 /; m + Condition z d z m z 1 2 c g 2 -1 m -1 k 0 m -1 2 -1 Binomial m k 1 2 d k -1 d m 3 2 -1 c 2 1 2 c 2 4 2 d k -1 d m -1 -1 g FresnelC -1 c 2 2 d k -1 d m z 1 2 2 d k -1 d m 1 2 2 1 2 -1 -1 c 2 1 2 FresnelS -1 c 2 2 d k -1 d m z 1 2 2 d k -1 d m 1 2 2 1 2 -1 c 2 4 2 d k -1 d m -1 -1 g -1 2 2 d k -1 d m 1 2 z 1 2 c g 2 d k -1 d m z 1 2 d k -1 d m 3 2 -1 -1 c 2 1 2 -1 c 2 4 2 d k -1 d m -1 g FresnelC c 2 2 d k -1 d m z 1 2 2 d k -1 d m 1 2 2 1 2 -1 -1 c 2 1 2 FresnelS c 2 2 d k -1 d m z 1 2 2 d k -1 d m 1 2 2 1 2 -1 -1 c 2 4 2 d k -1 d m -1 g 2 2 d k -1 d m 1 2 z 1 2 c g -1 2 d k -1 d m z -1 2 1 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 z 1 2 c g -1 c z 1 2 z 1 2 c g c 2 -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", 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["Cosh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["c", " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]], SuperscriptBox["c", "2"]]]], "+", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18