Complete elliptic integral of the first kind: Integration (formula 08.02.21.0019)

EllipticK

$Mathematica Notation$

For the direct function itself

http://functions.wolfram.com/08.02.21.0019.01

Input Form

Integrate[t^(\[Alpha] - 1) EllipticK[-t], {t, 0, Infinity}] == (Gamma[1/2 - \[Alpha]]^2 Gamma[\[Alpha]])/(2 Gamma[1 - \[Alpha]]) /; 0 < Re[\[Alpha]] < 1/2

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["EllipticK", "[", RowBox[List["-", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]"]], "]"]], "2"], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["1", "2"]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <real /> <ci> α </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["EllipticK", "[", RowBox[List["-", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Alpha]"]], "]"]], "2"], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["1", "2"]]]]]]]]]

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

2001-10-29