Exponential function: Integration (formula 01.03.21.0158)

Exp

$Mathematica Notation$

Elementary Functions

Indefinite integration

Involving one direct function and elementary functions

Arguments involving polynomials

Involving a z²+c

http://functions.wolfram.com/01.03.21.0158.01

Input Form

Integrate[d^(a z^2 + c), z] == (d^c Sqrt[Pi] Erfi[Sqrt[a] z Sqrt[Log[d]]])/ (2 Sqrt[a] Sqrt[Log[d]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["d", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["d", "c"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox["a"], " ", "z", " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]]]]]]]

MathML Form

<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> <msup> <mi> d </mi> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <msup> <mi> d </mi> <mi> c </mi> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <ci> d </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> d </ci> <ci> c </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> d </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> d </ci> </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[SuperscriptBox["d_", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", "c_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["d", "c"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox["a"], " ", "z", " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]]]]]]]

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

2002-12-18