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 Cosh

 http://functions.wolfram.com/01.20.21.0768.01

 Input Form

 Integrate[Cos[c z]^\[Mu] Cosh[a z], z] == ((1/(2 (a^2 + c^2 \[Mu]^2))) Cos[c z]^\[Mu] ((-a + I c \[Mu]) Hypergeometric2F1[(I (a + I c \[Mu]))/(2 c), -\[Mu], 1 + (I a)/(2 c) - \[Mu]/2, -E^(2 I c z)] + E^(2 a z) (a + I c \[Mu]) Hypergeometric2F1[-((I a + c \[Mu])/(2 c)), -\[Mu], (1/2) (2 - (I a)/c - \[Mu]), -E^(2 I c z)]))/ (E^(a z) (1 + E^(2 I c z))^\[Mu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", "z"]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "c"], "-", "\[Mu]"]], ")"]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]]]], ")"]], " "]]]]]]

 MathML Form

 cos μ ( c z ) cosh ( a z ) z 1 2 ( a 2 + c 2 μ 2 ) ( - a z ( 1 + 2 c z ) - μ cos μ ( c z ) ( ( c μ - a ) 2 F 1 ( ( a + c μ ) 2 c , - μ ; 1 + a 2 c - μ 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 a z ( a + c μ ) 2 F 1 ( - a + c μ 2 c , - μ ; 1 2 ( 2 - a c - μ ) ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "c"], "-", "\[Mu]"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) z c z μ a z 1 2 a 2 c 2 μ 2 -1 -1 a z 1 2 c z -1 μ c z μ c μ -1 a Hypergeometric2F1 a c μ 2 c -1 -1 μ 1 a 2 c -1 -1 μ 2 -1 -1 2 c z 2 a z a c μ Hypergeometric2F1 -1 a c μ 2 c -1 -1 μ 1 2 2 -1 a c -1 -1 μ -1 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["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", "z"]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "c"], "-", "\[Mu]"]], ")"]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["\[Mu]", "2"]]]]], ")"]]]]]]]]]

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

 2002-12-18