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 | | http://functions.wolfram.com/01.03.21.0681.01 | 
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 | | Integrate[z^n E^(b Sqrt[z] + e) (E^(c Sqrt[z] + g))^\[Nu], z] == 
  -(2 E^(e - c Sqrt[z] \[Nu]) (E^(g + c Sqrt[z]))^\[Nu] 
    (b + c \[Nu])^(-2 n - 2) Gamma[2 (1 + n), (-Sqrt[z]) (b + c \[Nu])]) /; 
 Element[n, Integers] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", "e"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]]], ")"]], "\[Nu]"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", RowBox[List["c", " ", SqrtBox["z"], " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", SqrtBox["z"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mrow>  <mo> ∫ </mo>  <mrow>  <msup>  <mi> z </mi>  <mi> n </mi>  </msup>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> + </mo>  <mi> e </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> ⁢ </mo>  <mi> c </mi>  </mrow>  <mo> + </mo>  <mi> g </mi>  </mrow>  </msup>  <mo> ) </mo>  </mrow>  <mi> ν </mi>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ⅆ </mo>  <mi> z </mi>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> e </mi>  <mo> - </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> ⁢ </mo>  <mi> ν </mi>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> ⁢ </mo>  <mi> c </mi>  </mrow>  <mo> + </mo>  <mi> g </mi>  </mrow>  </msup>  <mo> ) </mo>  </mrow>  <mi> ν </mi>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> ν </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  <mo> ⁢ </mo>  <mi> n </mi>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> n </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mo> - </mo>  <msqrt>  <mi> z </mi>  </msqrt>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> ν </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> n </mi>  <mo> ∈ </mo>  <semantics>  <mi> ℤ </mi>  <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation>  </semantics>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <int />  <bvar>  <ci> z </ci>  </bvar>  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <ci> n </ci>  </apply>  <apply>  <power />  <exponentiale />  <apply>  <plus />  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> b </ci>  </apply>  <ci> e </ci>  </apply>  </apply>  <apply>  <power />  <apply>  <power />  <exponentiale />  <apply>  <plus />  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> c </ci>  </apply>  <ci> g </ci>  </apply>  </apply>  <ci> ν </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -2 </cn>  <apply>  <power />  <exponentiale />  <apply>  <plus />  <ci> e </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <ci> c </ci>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> ν </ci>  </apply>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <power />  <exponentiale />  <apply>  <plus />  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> c </ci>  </apply>  <ci> g </ci>  </apply>  </apply>  <ci> ν </ci>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <ci> c </ci>  <ci> ν </ci>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -2 </cn>  <ci> n </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <ci> c </ci>  <ci> ν </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <in />  <ci> n </ci>  <integers />  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", "e_"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", RowBox[List["c", " ", SqrtBox["z"], " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", SqrtBox["z"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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