Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
 











Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric functions > Involving sin > Involving sin(b z)





http://functions.wolfram.com/01.22.21.0037.01









  


  










Input Form





Integrate[Sin[b z] Coth[c z], z] == (1/(2 (4 c^2 b + b^3))) (((4 c^2 + b^2) Hypergeometric2F1[-((I b)/(2 c)), 1, 1 - (I b)/(2 c), E^(2 c z)] + (4 c^2 + b^2) E^(2 I b z) Hypergeometric2F1[(I b)/(2 c), 1, 1 + (I b)/(2 c), E^(2 c z)] + b ((-2 I c + b) E^(2 c z) Hypergeometric2F1[1 - (I b)/(2 c), 1, 2 - (I b)/(2 c), E^(2 c z)] + (2 I c + b) E^(2 (c + I b) z) Hypergeometric2F1[1 + (I b)/(2 c), 1, 2 + (I b)/(2 c), E^(2 c z)]))/ E^(I b z))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"], " ", "b"]], "+", SuperscriptBox["b", "3"]]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "c"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "c"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "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> <mrow> <mi> sin </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> coth </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> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> <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> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </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;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;1&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&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> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </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> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </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;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&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> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mtext> </mtext> <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> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </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;1&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;+&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&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> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <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> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </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;1&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;b&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&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> <mo> ) </mo> </mrow> </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> <sin /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <coth /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </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[RowBox[List["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "c"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "c"]], "+", "b"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["c", "2"], " ", "b"]], "+", SuperscriptBox["b", "3"]]], ")"]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.