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http://functions.wolfram.com/01.03.21.0413.01
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Integrate[(E^(b Sqrt[z]))^\[Mu] (E^(Sqrt[z] c + f z + g))^\[Nu], z] ==
(1/(2 (f \[Nu])^(3/2))) (((E^(b Sqrt[z]))^\[Mu] (E^(g + c Sqrt[z] + f z))^
\[Nu] (2 E^((b \[Mu] + c \[Nu] + 2 f Sqrt[z] \[Nu])^2/(4 f \[Nu]))
Sqrt[f \[Nu]] - Sqrt[Pi] (b \[Mu] + c \[Nu])
Erfi[(b \[Mu] + c \[Nu] + 2 f Sqrt[z] \[Nu])/(2 Sqrt[f \[Nu]])]))/
E^((b \[Mu] + (c + 2 f Sqrt[z]) \[Nu])^2/(4 f \[Nu])))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SqrtBox["z"], " "]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[SqrtBox["z"], " ", "c"]], "+", RowBox[List["f", " ", "z"]], "+", "g"]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", "\[Nu]"]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SqrtBox["z"]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]], " ", SqrtBox[RowBox[List["f", " ", "\[Nu]"]]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["f", " ", "\[Nu]"]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> μ </mi> </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> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> g </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SqrtBox["z_"]]]], ")"]], "\[Mu]_"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[SqrtBox["z_"], " ", "c_"]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SqrtBox["z"]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]], " ", SqrtBox[RowBox[List["f", " ", "\[Nu]"]]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["f", " ", "\[Nu]"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", "\[Nu]"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
