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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric functions > Involving powers of cos > Involving cosm(b zr) cosh(c zr)





http://functions.wolfram.com/01.20.21.0800.01









  


  








Input Form





Integrate[Cos[b Sqrt[z]]^m Cosh[c Sqrt[z]], z] == -((1/c^2) (2^(1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) (Cosh[c Sqrt[z]] - c Sqrt[z] Sinh[c Sqrt[z]]))) + 2^(1 - m) Sum[Binomial[m, k] ((Cos[(I c + b (2 k - m)) Sqrt[z]] + (I c + b (2 k - m)) Sqrt[z] Sin[(I c + b (2 k - m)) Sqrt[z]])/ (I c + b (2 k - m))^2 + (Cos[((-I) c + b (2 k - m)) Sqrt[z]] + ((-I) c + b (2 k - m)) Sqrt[z] Sin[((-I) c + b (2 k - m)) Sqrt[z]])/ ((-I) c + b (2 k - m))^2), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0








Standard Form





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MathML Form







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</apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2002-12-18