Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cosh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.20.21.1271.01









  


  








Input Form





Integrate[E^(p Sqrt[z]) Cos[b Sqrt[z]] Cosh[c Sqrt[z]], z] == (1/2) (E^(((-I) b - c + p) Sqrt[z]) (1/(b - I c + I p)^2 + Sqrt[z]/((-I) b - c + p)) + E^((I b - c + p) Sqrt[z]) (1/(b + I (c - p))^2 + Sqrt[z]/(I b - c + p)) + E^(((-I) b + c + p) Sqrt[z]) (1/(b + I (c + p))^2 + Sqrt[z]/((-I) b + c + p)) + E^((I b + c + p) Sqrt[z]) (1/(b - I (c + p))^2 + Sqrt[z]/(I b + c + p)))








Standard Form





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








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> p </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mi> z </mi> </msqrt> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mi> z </mi> </msqrt> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mi> z </mi> </msqrt> <mrow> <mi> c </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mi> z </mi> </msqrt> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> p </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <ci> p </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SqrtBox["z_"]]]], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "+", "p"]], ")"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"]], "+", FractionBox[SqrtBox["z"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "+", "p"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "+", "p"]], ")"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]]]]]], ")"]], "2"]], "+", FractionBox[SqrtBox["z"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "+", "p"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", "p"]], ")"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]], ")"]], "2"]], "+", FractionBox[SqrtBox["z"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", "p"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", "p"]], ")"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]], ")"]], "2"]], "+", FractionBox[SqrtBox["z"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", "p"]]]]], ")"]]]]]], ")"]]]]]]]]


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





2002-12-18