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
 











Sec






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.11.21.0059.01









  


  










Input Form





Integrate[E^(p z) Cos[b z] Sec[c z], z] == -((I E^((I b + I c + p) z) Hypergeometric2F1[1, (b + c - I p)/(2 c), (b + 3 c - I p)/(2 c), -E^(2 I c z)])/(b + c - I p)) + (1/(b - c + I p)) (I E^(((-I) b + I c + p) z) Hypergeometric2F1[1, -((b - c + I p)/(2 c)), -((b - 3 c + I p)/(2 c)), -E^(2 I c z)])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["b", "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", "+", RowBox[List["3", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["b", "-", "c", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", FractionBox[RowBox[List["b", "-", "c", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", "-", RowBox[List["3", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]]]]]]










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> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, Hypergeometric2F1], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;b&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;p&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;b&quot;, &quot;-&quot;, RowBox[List[&quot;3&quot;, &quot; &quot;, &quot;c&quot;]], &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;p&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, Hypergeometric2F1], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;c&quot;, &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;p&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot;+&quot;, RowBox[List[&quot;3&quot;, &quot; &quot;, &quot;c&quot;]], &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;p&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> </mfrac> </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> <ci> z </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> p </ci> </apply> </apply> </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["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["b", "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List["b", "+", RowBox[List["3", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["-", FractionBox[RowBox[List["b", "-", "c", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", "-", RowBox[List["3", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "-", "c", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]]]]]]]]]










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





2002-10-15