|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.19.21.2443.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[(a + b z)^\[Beta] d^z Sinh[c z] Sinh[e z], z] ==
((1/4) (a + b z)^\[Beta]
((1/b) ((a + b z) Gamma[1 + \[Beta], -(((a + b z) (c - e + Log[d]))/b)]
(-(((a + b z) (c - e + Log[d]))/b))^(-1 - \[Beta])
(Cosh[(a (c - e))/b] - Sinh[(a (c - e))/b])) +
(1/b) ((a + b z) Gamma[1 + \[Beta], ((a + b z) (c - e - Log[d]))/b]
(((a + b z) (c - e - Log[d]))/b)^(-1 - \[Beta])
(Cosh[(a (c - e))/b] + Sinh[(a (c - e))/b])) +
(1/(c + e + Log[d]))
((Gamma[1 + \[Beta], -(((a + b z) (c + e + Log[d]))/b)]
(Cosh[(a (c + e))/b] - Sinh[(a (c + e))/b]))/
(-(((a + b z) (c + e + Log[d]))/b))^\[Beta]) -
(1/(c + e - Log[d])) ((Gamma[1 + \[Beta], ((a + b z) (c + e - Log[d]))/b]
(Cosh[(a (c + e))/b] + Sinh[(a (c + e))/b]))/
(((a + b z) (c + e - Log[d]))/b)^\[Beta])))/d^(a/b)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], "\[Beta]"], SuperscriptBox["d", "z"], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["d", RowBox[List["-", FractionBox["a", "b"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]], "-", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]], "+", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]], "-", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]], "+", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> <mo> ⁢ </mo> <msup> <mi> d </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> d </mi> <mrow> <mo> - </mo> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> e </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </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 /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> β </ci> </apply> <apply> <power /> <ci> d </ci> <ci> z </ci> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> β </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <ci> e </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <ci> e </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <ci> e </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> c </ci> <ci> e </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> d </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", "z_"]]]], ")"]], "\[Beta]_"], " ", SuperscriptBox["d_", "z_"], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["d", RowBox[List["-", FractionBox["a", "b"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]], "-", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], "b"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]], "+", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "-", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], "b"], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]], "-", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], RowBox[List["c", "+", "e", "+", RowBox[List["Log", "[", "d", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]], ")"]]]], "b"], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]], "+", RowBox[List["Sinh", "[", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["c", "+", "e"]], ")"]]]], "b"], "]"]]]], ")"]]]], RowBox[List["c", "+", "e", "-", RowBox[List["Log", "[", "d", "]"]]]]]]], ")"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|