|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.16.21.0010.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[E^(a z) ArcCot[b z], z] == (E^(a z) ArcCot[b z])/a -
(I E^((I a)/b) ExpIntegralEi[-((I a)/b) + a z])/(2 a) +
(I ExpIntegralEi[(I a)/b + a z])/(2 (a E^((I a)/b)))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["ArcCot", "[", RowBox[List["b", " ", "z"]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["ArcCot", "[", RowBox[List["b", " ", "z"]], "]"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], RowBox[List["2", " ", "a"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "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> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </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> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mi> a </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mtext> </mtext> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mi> b </mi> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mi> b </mi> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mi> b </mi> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <arccot /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccot /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </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["\[ExponentialE]", RowBox[List["a_", " ", "z_"]]], " ", RowBox[List["ArcCot", "[", RowBox[List["b_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["ArcCot", "[", RowBox[List["b", " ", "z"]], "]"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], RowBox[List["2", " ", "a"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"], "+", RowBox[List["a", " ", "z"]]]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "b"]]]], ")"]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|