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
 











Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function > Involving exp > Involving ab zr+e cos(f z+g)





http://functions.wolfram.com/01.07.21.0209.01









  


  










Input Form





Integrate[a^(b Sqrt[z] + e) Cos[f z + g], z] == ((-(1/4)) a^e ((2 I a^(b Sqrt[z]) (-(1/f) + E^(2 I (g + f z))/f))/ E^(I f z) + (b Sqrt[Pi] Erfi[(-2 I f Sqrt[z] + b Log[a])/ (2 Sqrt[(-I) f])] Log[a])/(E^((I b^2 Log[a]^2)/(4 f)) ((-I) f)^(3/2)) + (b E^(2 I g + (I b^2 Log[a]^2)/(4 f)) Sqrt[Pi] Erfi[(2 I f Sqrt[z] + b Log[a])/(2 Sqrt[I f])] Log[a])/(I f)^(3/2)))/ E^(I g)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", "e"]]], RowBox[List["Cos", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "e"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", RowBox[List["b", " ", SqrtBox["z"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "f"]]], "+", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]], ")"]]]]], "f"]]], ")"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "f"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "f"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]










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> a </mi> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> e </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> g </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mi> e </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> g </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> g </mi> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mi> f </mi> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> f </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> f </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> g </mi> <mtext> </mtext> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> </apply> <ci> e </ci> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <ci> a </ci> <ci> e </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> g </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> f </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <plus /> <ci> g </ci> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> f </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> f </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <imaginaryi /> <apply> <power /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> f </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <ci> f </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> f </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> f </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> f </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> g </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> f </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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["a_", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", "e_"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "e"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", RowBox[List["b", " ", SqrtBox["z"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "f"]]], "+", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]], ")"]]]]], "f"]]], ")"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "f"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "g"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "f"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]










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





2002-12-18