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Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving products of powers of the direct function > Involving product of power of the direct function and the direct function > Involving eb zr+d z+e (ec zr)nu





http://functions.wolfram.com/01.03.21.0338.01









  


  










Input Form





Integrate[E^(b z^2 + d z + e) (E^(c z^2))^\[Nu], z] == (E^(e + z (d + b z) - (d + 2 b z + 2 c z \[Nu])^2/(4 (b + c \[Nu]))) (E^(c z^2))^\[Nu] Sqrt[Pi] Erfi[(d + 2 b z + 2 c z \[Nu])/ (2 Sqrt[b + c \[Nu]])])/(2 Sqrt[b + c \[Nu]])










Standard Form





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MathML Form







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</mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> e </mi> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <ci> e </ci> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> e </ci> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["b", " ", "z"]]]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", SuperscriptBox["z", "2"]]]], ")"]], "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z", " ", "\[Nu]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]]]]]]]]]]










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





2002-12-18