Gamma[z] == E^Integrate[(x^z - z (x - 1) - 1)/((x - 1) Log[x]), {x, 0, 1}] /; Re[z] > 0

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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> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <msup> <mi> &#8519; </mi> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mi> z </mi> </msup> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> x </mi> </mrow> </mrow> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <power /> <exponentiale /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ln /> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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A. Radović

2001-12-21