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/01.20.21.1532.01

 Input Form

 Integrate[Sinh[c z]^m Cosh[a z], z] == (I^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[a z])/(2^m a) + 2^(-1 - m) Sum[(-1)^(k + m) ((-E^((a + 2 c k - c m) z)) (a - 2 c k + c m) (a - c (-2 k + m)) + (-a - 2 c k + c m) ((-E^(I m Pi - (a - c (-2 k + m)) z)) (a - 2 c k + c m) - E^((-a) z + 2 c k z - c m z) (a - c (-2 k + m)) + E^(I m Pi + a z - 2 c k z + c m z) (a - c (-2 k + m)))) (Binomial[m, k]/((-a - 2 c k + c m) (a - 2 c k + c m) (a - c (-2 k + 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["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "a"], "+", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "k", " ", "z"]], "-", RowBox[List["c", " ", "m", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["a", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "k", " ", "z"]], "+", RowBox[List["c", " ", "m", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sinh m ( c z ) cosh ( a z ) z ( 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]]]]] m 2 - m ( 1 - m mod 2 \$CellContext`m 2 ) sinh ( a z ) a + 2 - m - 1 k = 0 m - 1 2 ( - 1 ) k + m ( ( - a - 2 c k + c m ) ( - m π - ( a - c ( m - 2 k ) ) z ( a - 2 c k + c m ) + c z m + π m + a z - 2 c k z ( a - c ( m - 2 k ) ) - - a z + 2 c k z - c m z ( a - c ( m - 2 k ) ) ) - ( a + 2 c k - c m ) z ( a - 2 c k + c m ) ( a - c ( m - 2 k ) ) ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] / ( ( - a - 2 c k + c m ) ( a - 2 c k + c m ) ( a - c ( m - 2 k ) ) ) /; m + Condition z c z m a z Binomial m m 2 -1 m 2 -1 m 1 -1 \$CellContext`m 2 a z a -1 2 -1 m -1 k 0 m -1 2 -1 -1 k m -1 a -1 2 c k c m -1 m -1 a -1 c m -1 2 k z a -1 2 c k c m c z m m a z -1 2 c k z a -1 c m -1 2 k -1 -1 a z 2 c k z -1 c m z a -1 c m -1 2 k -1 a 2 c k -1 c m z a -1 2 c k c m a -1 c m -1 2 k Binomial m k -1 a -1 2 c k c m a -1 2 c k c m a -1 c m -1 2 k -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "a"], "+", 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"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "k", " ", "z"]], "-", RowBox[List["c", " ", "m", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["a", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "k", " ", "z"]], "+", RowBox[List["c", " ", "m", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "c", " ", "k"]], "+", RowBox[List["c", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18