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; }

 Csch

 http://functions.wolfram.com/01.23.21.0212.01

 Input Form

 Integrate[Cos[a z]^m Cosh[c z]^\[Mu] Csch[c z], z] == ((1/(c (\[Mu] - 1))) 2^(1 - m) Binomial[m, m/2] Cosh[c z]^\[Mu] (1 - Mod[m, 2]) AppellF1[-((\[Mu] - 1)/2), -\[Mu], 1, (3 - \[Mu])/2, -E^(-2 c z), E^(-2 c z)])/(E^(c z) (1 + E^(-2 c z))^\[Mu]) + (2^(1 - m) Cosh[c z]^\[Mu] Sum[((E^(I a (m - 2 s) z) AppellF1[-((I a m - 2 I a s + c \[Mu] - c)/ (2 c)), -\[Mu], 1, -((I a m - 2 I a s + c (-3 + \[Mu]))/(2 c)), -E^(-2 c z), E^(-2 c z)])/(I a (m - 2 s) + c \[Mu] - c) + (E^(((-I) a m + 2 I a s) z) AppellF1[(I a m - 2 I a s - c \[Mu] + c)/ (2 c), -\[Mu], 1, (I a m - 2 I a s + c (3 - \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)])/((-I) a m + 2 I a s + c \[Mu] - c)) Binomial[m, s], {s, 0, Floor[(1/2) (-1 + m)]}])/ (E^(c z) (1 + E^(-2 c z))^\[Mu]) /; Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["c", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]], SuperscriptBox["2", RowBox[List["1", "-", "m"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Mu]", "-", "1"]], RowBox[List["2", " "]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", FractionBox[RowBox[List["3", "-", "\[Mu]"]], "2"], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], " ", "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "-", RowBox[List["c", " ", "\[Mu]"]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]], ")"]]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( a z ) cosh μ ( c z ) csch ( c z ) z 2 1 - m cosh μ ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 + - 2 c z ) - μ c ( μ - 1 ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] - c z F 1 AppellF1 ( - 1 2 ( μ - 1 ) ; - μ , 1 ; 3 - μ 2 ; - - 2 c z , - 2 c z ) + 2 1 - m - c z cosh μ ( c z ) ( 1 + - 2 c z ) - μ s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( 2 a s - a m ) z μ c - c - a m + 2 a s F 1 AppellF1 ( - μ c + c + a m - 2 a s 2 c ; - μ , 1 ; a m - 2 a s + c ( 3 - μ ) 2 c ; - - 2 c z , - 2 c z ) + a ( m - 2 s ) z μ c - c + a ( m - 2 s ) F 1 AppellF1 ( - μ c - c + a m - 2 a s 2 c ; - μ , 1 ; - a m - 2 a s + c ( μ - 3 ) 2 c ; - - 2 c z , - 2 c z ) ) z a z m c z μ c z 2 1 -1 m c z μ 1 -1 \$CellContext`m 2 1 -2 c z -1 μ c μ -1 -1 Binomial m m 2 -1 -1 c z AppellF1 -1 1 2 μ -1 -1 μ 1 3 -1 μ 2 -1 -1 -2 c z -2 c z 2 1 -1 m -1 c z c z μ 1 -2 c z -1 μ s 0 m -1 2 -1 Binomial m s 2 a s -1 a m z μ c -1 c -1 a m 2 a s -1 AppellF1 -1 μ c c a m -1 2 a s 2 c -1 -1 μ 1 a m -1 2 a s c 3 -1 μ 2 c -1 -1 -2 c z -2 c z a m -1 2 s z μ c -1 c a m -1 2 s -1 AppellF1 -1 μ c -1 c a m -1 2 a s 2 c -1 -1 μ 1 -1 a m -1 2 a s c μ -3 2 c -1 -1 -2 c z -2 c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", FractionBox[RowBox[List["3", "-", "\[Mu]"]], "2"], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "1"]], ")"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "-", RowBox[List["c", " ", "\[Mu]"]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "s"]], "+", RowBox[List["c", " ", "\[Mu]"]], "-", "c"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18