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 cos > Involving cos(b z) sinh(c zr)





http://functions.wolfram.com/01.19.21.0715.01









  


  










Input Form





Integrate[Cos[b z] Sinh[c z^2], z] == (1/2) I Sqrt[Pi/2] ((1/Sqrt[(-I) c]) ((-Cosh[b^2/(4 c)]) FresnelS[(b + 2 I c z)/ (Sqrt[(-I) c] Sqrt[2 Pi])] + I FresnelC[(b + 2 I c z)/(Sqrt[(-I) c] Sqrt[2 Pi])] Sinh[b^2/(4 c)]) - (1/Sqrt[I c]) (Cosh[b^2/(4 c)] FresnelS[(-b + 2 I c z)/ (Sqrt[I c] Sqrt[2 Pi])] + I FresnelC[(-b + 2 I c z)/(Sqrt[I c] Sqrt[2 Pi])] Sinh[b^2/(4 c)]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Sinh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cosh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]










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> 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> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> b </mi> </mrow> <mrow> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> b </mi> </mrow> <mrow> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cosh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cosh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], ")"]]]]]]]]










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





2002-12-18