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 Coth

 http://functions.wolfram.com/01.22.21.0185.01

 Input Form

 Integrate[Cos[a z]^m Cosh[c z]^\[Mu] Coth[c z], z] == (1/(c \[Mu])) ((Cosh[c z]^\[Mu] AppellF1[-(\[Mu]/2), 1, -1 - \[Mu], 1 - \[Mu]/2, E^(2 c z), -E^(2 c z)] Binomial[m, m/2] (1 - Mod[m, 2]))/ (2^m (1 + E^(2 c z))^\[Mu])) - (Cosh[c z]^\[Mu] Sum[Binomial[m, k] (-(AppellF1[((-I) a (-2 k + m) - c \[Mu])/(2 c), 1, -\[Mu] - 1, (1/2) (2 + ((-I) a (-2 k + m))/c - \[Mu]), E^(2 c z), -E^(2 c z)]/ E^(I a (-2 k + m) z))/(I a (-2 k + m) + c \[Mu]) + (E^(I a (-2 k + m) z) AppellF1[(I a (-2 k + m) - c \[Mu])/(2 c), 1, -\[Mu] - 1, (1/2) (2 + (I a (-2 k + m))/c - \[Mu]), E^(2 c z), -E^(2 c z)])/(I a (-2 k + m) - c \[Mu])), {k, 0, Floor[(1/2) (-1 + m)]}])/(2^m (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["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["c", " ", "\[Mu]"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "1", ",", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], 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[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]], RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18