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LogIntegral






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > LogIntegral[z] > Series representations > Generalized power series > Expansions at z==1 > For the function itself





http://functions.wolfram.com/06.36.06.0016.01









  


  










Input Form





LogIntegral[z] == (1/2) (Log[z - 1] - Log[1/(z - 1)]) + EulerGamma + Sum[((-1)^j/(j j!)) Sum[Subscript[p, h, j] (z - 1)^(h + j), {h, 0, Infinity}], {j, 1, Infinity}] + Sum[((-1)^(j - 1)/j) Sum[Subscript[q, k, j] (z - 1)^(k + j), {k, 0, Infinity}], {j, 1, Infinity}] /; Subscript[p, h, j] == (1/h) Sum[(((-1)^i (j i - h + i))/(i + 1)) Subscript[p, h - i, j], {i, 1, h}] && Subscript[p, 0, j] == (-1)^j && Subscript[q, k, j] == (2/k) Sum[(((-1)^i (j i - k + i))/(i + 2)) Subscript[q, k - i, j], {i, 1, k}] && Subscript[q, 0, j] == (-(1/2))^j










Standard Form





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MathML Form







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</mo> <mrow> <msub> <mi> q </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> j </mi> </mrow> </msub> <mo> &#10869; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LogIntegral </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <eulergamma /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <ci> j </ci> <apply> <factorial /> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> h </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> h </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> h </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> h </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> h </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> h </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> h </ci> </apply> <ci> i </ci> <apply> <times /> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> h </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 0 </cn> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <plus /> <apply> <times /> <ci> j </ci> <ci> i </ci> </apply> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> q </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> q </ci> <cn type='integer'> 0 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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