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/01.20.21.0664.01

 Input Form

 Integrate[Sin[b z^2 + d z + e]^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] ((Cos[e (2 k - m) - ((-I) f + 2 d k - d m)^2/(4 ((-I) c + 2 b k - b m)) - (m Pi)/2] FresnelC[((-I) f + 2 d k - d m + 2 ((-I) c + 2 b k - b m) z)/(Sqrt[(-I) c + 2 b k - b m] Sqrt[2 Pi])] - FresnelS[((-I) f + 2 d k - d m + 2 ((-I) c + 2 b k - b m) z)/(Sqrt[(-I) c + 2 b k - b m] Sqrt[2 Pi])] Sin[e (2 k - m) - ((-I) f + 2 d k - d m)^2/ (4 ((-I) c + 2 b k - b m)) - (m Pi)/2])/ Sqrt[(-I) c + 2 b k - b m] + (Cos[(-e) (2 k - m) + (I f + 2 d k - d m)^2/(4 (I c + 2 b k - b m)) + (m Pi)/2] FresnelC[(I f + 2 d k - d m + 2 (I c + 2 b k - b m) z)/ (Sqrt[I c + 2 b k - b m] Sqrt[2 Pi])] + FresnelS[(I f + 2 d k - d m + 2 (I c + 2 b k - b m) z)/ (Sqrt[I c + 2 b k - b m] Sqrt[2 Pi])] Sin[(-e) (2 k - m) + (I f + 2 d k - d m)^2/(4 (I c + 2 b k - b m)) + (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

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RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]]]]], "+", FractionBox[RowBox[List["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 + d z + e ) 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 + 2 b k - b m ( cos ( ( f + 2 d k - d m ) 2 4 ( c + 2 b k - b m ) - e ( 2 k - m ) + m π 2 ) C TagBox["C", FresnelC] ( f + 2 d k - d m + 2 ( c + 2 b k - b m ) z c + 2 b k - b m 2 π ) + S TagBox["S", FresnelS] ( f + 2 d k - d m + 2 ( c + 2 b k - b m ) z c + 2 b k - b m 2 π ) sin ( ( f + 2 d k - d m ) 2 4 ( c + 2 b k - b m ) - e ( 2 k - m ) + m π 2 ) ) + 1 - c + 2 b k - b m ( cos ( - ( - f + 2 d k - d m ) 2 4 ( - c + 2 b k - b m ) + e ( 2 k - m ) - m π 2 ) C TagBox["C", FresnelC] ( - f + 2 d k - d m + 2 ( - c + 2 b k - b m ) z - c + 2 b k - b m 2 π ) - S TagBox["S", FresnelS] ( - f + 2 d k - d m + 2 ( - c + 2 b k - b m ) z - c + 2 b k - b m 2 π ) sin ( - ( - f + 2 d k - d m ) 2 4 ( - c + 2 b k - b m ) + e ( 2 k - m ) - m π 2 ) ) ) /; m + Condition z b z 2 d z e 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 c 2 b k -1 b m 1 2 -1 f 2 d k -1 d m 2 4 c 2 b k -1 b m -1 -1 e 2 k -1 m m 2 -1 FresnelC f 2 d k -1 d m 2 c 2 b k -1 b m z c 2 b k -1 b m 1 2 2 1 2 -1 FresnelS f 2 d k -1 d m 2 c 2 b k -1 b m z c 2 b k -1 b m 1 2 2 1 2 -1 f 2 d k -1 d m 2 4 c 2 b k -1 b m -1 -1 e 2 k -1 m m 2 -1 1 -1 c 2 b k -1 b m 1 2 -1 -1 -1 f 2 d k -1 d m 2 4 -1 c 2 b k -1 b m -1 e 2 k -1 m -1 m 2 -1 FresnelC -1 f 2 d k -1 d m 2 -1 c 2 b k -1 b m z -1 c 2 b k -1 b m 1 2 2 1 2 -1 -1 FresnelS -1 f 2 d k -1 d m 2 -1 c 2 b k -1 b m z -1 c 2 b k -1 b m 1 2 2 1 2 -1 -1 -1 f 2 d k -1 d m 2 4 -1 c 2 b k -1 b m -1 e 2 k -1 m -1 m 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18