Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Power




Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Representations through more general functions > Through Meijer G > Classical cases involving sqrt in the arguments and unit step theta





http://functions.wolfram.com/01.02.26.0026.01









  


  








Input Form





UnitStep[1 - Abs[z]] ((Sqrt[1 + Sqrt[z]] - Sqrt[1 - Sqrt[z]])^\[Beta] - (Sqrt[1 + Sqrt[z]] + Sqrt[1 - Sqrt[z]])^\[Beta]) == (-2^(\[Beta]/2 - 1)) \[Beta] Sqrt[Pi] MeijerG[{{}, {(\[Beta] + 2)/4, 1 + \[Beta]/4}}, {{0, \[Beta]/2}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SqrtBox["z"]]]], "-", SqrtBox[RowBox[List["1", "-", SqrtBox["z"]]]]]], ")"]], "\[Beta]"], "-", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SqrtBox["z"]]]], "+", SqrtBox[RowBox[List["1", "-", SqrtBox["z"]]]]]], ")"]], "\[Beta]"]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[FractionBox["\[Beta]", "2"], "-", "1"]]]]], "\[Beta]", SqrtBox["\[Pi]"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Beta]", "+", "2"]], "4"], ",", RowBox[List["1", "+", FractionBox["\[Beta]", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["\[Beta]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]








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> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mfrac> <mi> &#946; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#946; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> &#946; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mi> &#946; </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Beta]&quot;, &quot;+&quot;, &quot;2&quot;]], &quot;4&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Beta]&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Beta]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mfrac> <mi> &#946; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#946; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> &#946; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mi> &#946; </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Beta]&quot;, &quot;+&quot;, &quot;2&quot;]], &quot;4&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Beta]&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;\[Beta]&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z_", "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SqrtBox["z_"]]]], "-", SqrtBox[RowBox[List["1", "-", SqrtBox["z_"]]]]]], ")"]], "\[Beta]_"], "-", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", SqrtBox["z_"]]]], "+", SqrtBox[RowBox[List["1", "-", SqrtBox["z_"]]]]]], ")"]], "\[Beta]_"]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[FractionBox["\[Beta]", "2"], "-", "1"]]]]], " ", "\[Beta]", " ", SqrtBox["\[Pi]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Beta]", "+", "2"]], "4"], ",", RowBox[List["1", "+", FractionBox["\[Beta]", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["\[Beta]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29