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
 











Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric functions > Involving algebraic functions of sin and cos > Involving sinh(d z)(a sin(e z)+b cos(e z))beta





http://functions.wolfram.com/01.19.21.0870.01









  


  










Input Form





Integrate[(a Sin[e z] + b Cos[e z])^\[Beta] Sinh[d z], z] == (2^(-1 - \[Beta]) (((-I) a (-1 + E^(2 I e z)) + b (1 + E^(2 I e z)))/ E^(I e z))^\[Beta] (E^(d z) (d + I e \[Beta]) Hypergeometric2F1[-((I (d - I e \[Beta]))/(2 e)), -\[Beta], (1/2) (2 - (I d)/e - \[Beta]), ((a + I b) E^(2 I e z))/(a - I b)] + ((d - I e \[Beta]) Hypergeometric2F1[(I (d + I e \[Beta]))/(2 e), -\[Beta], (1/2) (2 + (I d)/e - \[Beta]), ((a + I b) E^(2 I e z))/ (a - I b)])/E^(d z)))/(1 + (((-I) a + b) E^(2 I e z))/(I a + b))^ \[Beta]/((d - I e \[Beta]) (d + I e \[Beta]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], ")"]]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]], RowBox[List["2", " ", "e"]]]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d"]], "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]], RowBox[List["2", " ", "e"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d"]], "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]], ")"]]]]]]]]










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> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </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> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mtext> </mtext> </mrow> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mi> e </mi> </mfrac> <mo> - </mo> <mi> &#946; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </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[FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;\[Beta]&quot;]]]], &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;\[Beta]&quot;]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;e&quot;], &quot;-&quot;, &quot;\[Beta]&quot;]], &quot;)&quot;]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]]]], &quot; &quot;, &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mi> e </mi> </mfrac> <mo> - </mo> <mi> &#946; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </mfrac> </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[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;d&quot;, &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;\[Beta]&quot;]]]], &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;\[Beta]&quot;]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;d&quot;]], &quot;e&quot;], &quot;-&quot;, &quot;\[Beta]&quot;]], &quot;)&quot;]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]]]], &quot; &quot;, &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </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 /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <sinh /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> d </ci> <apply> <power /> <ci> e </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]], ")"]], "\[Beta]_"], " ", RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], ")"]]]]]], ")"]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]], RowBox[List["2", " ", "e"]]]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d"]], "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]], RowBox[List["2", " ", "e"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d"]], "e"], "-", "\[Beta]"]], ")"]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "\[Beta]"]]]], ")"]]]]]]]]]










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





2002-12-18