Finding discrete solutions to inequality involving Exponential Integral












1












$begingroup$


I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that



$$-c text{Ei}left(-e^{frac{a-d}{c}} (n+1)right)+a-b (n+1)+c log (n+1)+gamma c < 0,$$



where $text{Ei}$ is the exponential integral, $a,b, c, d$ are arbitrary real constants, and $gamma$ is the Euler-Mascheroni constant.



I have tried moving the terms around etc., to no avail; Mathematica also does not seem able to solve this.



I was wondering if there are any easy ways to solve this, or at least simplify it?






inequality optimization integral-inequality discrete-optimization functional-inequalities







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that



    $$-c text{Ei}left(-e^{frac{a-d}{c}} (n+1)right)+a-b (n+1)+c log (n+1)+gamma c < 0,$$



    where $text{Ei}$ is the exponential integral, $a,b, c, d$ are arbitrary real constants, and $gamma$ is the Euler-Mascheroni constant.



    I have tried moving the terms around etc., to no avail; Mathematica also does not seem able to solve this.



    I was wondering if there are any easy ways to solve this, or at least simplify it?






    inequality optimization integral-inequality discrete-optimization functional-inequalities







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that



      $$-c text{Ei}left(-e^{frac{a-d}{c}} (n+1)right)+a-b (n+1)+c log (n+1)+gamma c < 0,$$



      where $text{Ei}$ is the exponential integral, $a,b, c, d$ are arbitrary real constants, and $gamma$ is the Euler-Mascheroni constant.



      I have tried moving the terms around etc., to no avail; Mathematica also does not seem able to solve this.



      I was wondering if there are any easy ways to solve this, or at least simplify it?






      inequality optimization integral-inequality discrete-optimization functional-inequalities







      share|cite|improve this question









      $endgroup$




      I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that



      $$-c text{Ei}left(-e^{frac{a-d}{c}} (n+1)right)+a-b (n+1)+c log (n+1)+gamma c < 0,$$



      where $text{Ei}$ is the exponential integral, $a,b, c, d$ are arbitrary real constants, and $gamma$ is the Euler-Mascheroni constant.



      I have tried moving the terms around etc., to no avail; Mathematica also does not seem able to solve this.



      I was wondering if there are any easy ways to solve this, or at least simplify it?






      inequality optimization integral-inequality discrete-optimization functional-inequalities




      inequality optimization integral-inequality discrete-optimization functional-inequalities






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 1 at 16:45









      jackson5jackson5

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