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/01.20.21.0825.01

 Input Form

 Integrate[Cos[d z + e]^m Cosh[c Sqrt[z] + f z], z] == (1/(I f)^(3/2)) (2^(-1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) (c Sqrt[2 Pi] (Cosh[c^2/(4 f)] FresnelC[(c + 2 f Sqrt[z])/ (Sqrt[I f] Sqrt[2 Pi])] - I FresnelS[(c + 2 f Sqrt[z])/ (Sqrt[I f] Sqrt[2 Pi])] Sinh[c^2/(4 f)]) + 2 I Sqrt[I f] Sinh[c Sqrt[z] + f z])) + 2^(-1 - m) Sum[Binomial[m, k] ((I c Sqrt[2 Pi] Cos[2 e k - e m + c^2/(4 ((-I) f + 2 d k - d m))] FresnelC[((-I) c + 2 ((-I) f + 2 d k - d m) Sqrt[z])/ (Sqrt[(-I) f + 2 d k - d m] Sqrt[2 Pi])] - I c Sqrt[2 Pi] FresnelS[((-I) c + 2 ((-I) f + 2 d k - d m) Sqrt[z])/ (Sqrt[(-I) f + 2 d k - d m] Sqrt[2 Pi])] Sin[2 e k - e m + c^2/(4 ((-I) f + 2 d k - d m))] - 2 I Sqrt[(-I) f + 2 d k - d m] Sinh[2 I e k - I e m + c Sqrt[z] + I ((-I) f + 2 d k - d m) z])/((-I) f + 2 d k - d m)^(3/2) + ((-I) c Sqrt[2 Pi] Cosh[-2 I e k + I e m - (I c^2)/ (4 (I f + 2 d k - d m))] FresnelC[ (I c + 2 (I f + 2 d k - d m) Sqrt[z])/(Sqrt[I f + 2 d k - d m] Sqrt[2 Pi])] - c Sqrt[2 Pi] FresnelS[ (I c + 2 (I f + 2 d k - d m) Sqrt[z])/(Sqrt[I f + 2 d k - d m] Sqrt[2 Pi])] Sinh[-2 I e k + I e m - (I c^2)/ (4 (I f + 2 d k - d m))] + 2 I Sqrt[I f + 2 d k - d m] Sinh[-2 I e k + I e m + c Sqrt[z] - I (I f + 2 d k - d m) z])/ (I f + 2 d k - d m)^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18