Sharp bounds for the principal branch of the Lambert W function?












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I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x in [1, infty)$. This far I've found this paper: On certain inequalities involving the Lambert W function, but I would like something that is sharper than their theorem $3.5$ in the range $1 leq x leq 3$. As an added complication, I want to be able to symbolically write down the integral of an expression involving this approximation, so I'd like it to be as simple as possible. I realise I might be looking for unicorns, but does anyone know of any such bounds?






reference-request asymptotics lambert-w







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    3












    $begingroup$


    I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x in [1, infty)$. This far I've found this paper: On certain inequalities involving the Lambert W function, but I would like something that is sharper than their theorem $3.5$ in the range $1 leq x leq 3$. As an added complication, I want to be able to symbolically write down the integral of an expression involving this approximation, so I'd like it to be as simple as possible. I realise I might be looking for unicorns, but does anyone know of any such bounds?






    reference-request asymptotics lambert-w







    share|cite|improve this question









    $endgroup$















      3












      3








      3


      1



      $begingroup$


      I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x in [1, infty)$. This far I've found this paper: On certain inequalities involving the Lambert W function, but I would like something that is sharper than their theorem $3.5$ in the range $1 leq x leq 3$. As an added complication, I want to be able to symbolically write down the integral of an expression involving this approximation, so I'd like it to be as simple as possible. I realise I might be looking for unicorns, but does anyone know of any such bounds?






      reference-request asymptotics lambert-w







      share|cite|improve this question









      $endgroup$




      I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x in [1, infty)$. This far I've found this paper: On certain inequalities involving the Lambert W function, but I would like something that is sharper than their theorem $3.5$ in the range $1 leq x leq 3$. As an added complication, I want to be able to symbolically write down the integral of an expression involving this approximation, so I'd like it to be as simple as possible. I realise I might be looking for unicorns, but does anyone know of any such bounds?






      reference-request asymptotics lambert-w




      reference-request asymptotics lambert-w






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      share|cite|improve this question











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      asked Jan 30 at 9:53









      JohannaJohanna

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      4,74541640






















          1 Answer
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          $begingroup$

          This is not a full answer.



          I had to work a similar problem years ago and I used two different series expansions.



          Built at $x=-frac 1e$
          $$W(x)=-1+p-frac{p^2}{3}+frac{11 p^3}{72}-frac{43 p^4}{540}+frac{769
          p^5}{17280}-frac{221 p^6}{8505}+frac{680863 p^7}{43545600}-frac{1963
          p^8}{204120}+frac{226287557 p^9}{37623398400}-frac{5776369
          p^{10}}{1515591000}+frac{169709463197 p^{11}}{69528040243200}-frac{1118511313
          p^{12}}{709296588000}+Oleft(p^{13}right)$$           in which $p= sqrt{(2(1+e x)}$



          Built at $x=0$
          $$W(x)=x-x^2+frac{3 x^3}{2}-frac{8 x^4}{3}+frac{125 x^5}{24}-frac{54
          x^6}{5}+frac{16807 x^7}{720}-frac{16384 x^8}{315}+frac{531441
          x^9}{4480}-frac{156250 x^{10}}{567}+frac{2357947691
          x^{11}}{3628800}-frac{2985984 x^{12}}{1925}+Oleft(x^{13}right)$$           For sure, you can truncate or extend these series.



          Numerical calculations show that the first one would be used for $-frac1 {e} leq x leq -frac1 {2e} $ and the second one for $-frac1 {2e} leq x leq 0 $.



          In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.






          share|cite|improve this answer









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            1 Answer
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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            This is not a full answer.



            I had to work a similar problem years ago and I used two different series expansions.



            Built at $x=-frac 1e$
            $$W(x)=-1+p-frac{p^2}{3}+frac{11 p^3}{72}-frac{43 p^4}{540}+frac{769
            p^5}{17280}-frac{221 p^6}{8505}+frac{680863 p^7}{43545600}-frac{1963
            p^8}{204120}+frac{226287557 p^9}{37623398400}-frac{5776369
            p^{10}}{1515591000}+frac{169709463197 p^{11}}{69528040243200}-frac{1118511313
            p^{12}}{709296588000}+Oleft(p^{13}right)$$             in which $p= sqrt{(2(1+e x)}$



            Built at $x=0$
            $$W(x)=x-x^2+frac{3 x^3}{2}-frac{8 x^4}{3}+frac{125 x^5}{24}-frac{54
            x^6}{5}+frac{16807 x^7}{720}-frac{16384 x^8}{315}+frac{531441
            x^9}{4480}-frac{156250 x^{10}}{567}+frac{2357947691
            x^{11}}{3628800}-frac{2985984 x^{12}}{1925}+Oleft(x^{13}right)$$             For sure, you can truncate or extend these series.



            Numerical calculations show that the first one would be used for $-frac1 {e} leq x leq -frac1 {2e} $ and the second one for $-frac1 {2e} leq x leq 0 $.



            In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              This is not a full answer.



              I had to work a similar problem years ago and I used two different series expansions.



              Built at $x=-frac 1e$
              $$W(x)=-1+p-frac{p^2}{3}+frac{11 p^3}{72}-frac{43 p^4}{540}+frac{769
              p^5}{17280}-frac{221 p^6}{8505}+frac{680863 p^7}{43545600}-frac{1963
              p^8}{204120}+frac{226287557 p^9}{37623398400}-frac{5776369
              p^{10}}{1515591000}+frac{169709463197 p^{11}}{69528040243200}-frac{1118511313
              p^{12}}{709296588000}+Oleft(p^{13}right)$$               in which $p= sqrt{(2(1+e x)}$



              Built at $x=0$
              $$W(x)=x-x^2+frac{3 x^3}{2}-frac{8 x^4}{3}+frac{125 x^5}{24}-frac{54
              x^6}{5}+frac{16807 x^7}{720}-frac{16384 x^8}{315}+frac{531441
              x^9}{4480}-frac{156250 x^{10}}{567}+frac{2357947691
              x^{11}}{3628800}-frac{2985984 x^{12}}{1925}+Oleft(x^{13}right)$$               For sure, you can truncate or extend these series.



              Numerical calculations show that the first one would be used for $-frac1 {e} leq x leq -frac1 {2e} $ and the second one for $-frac1 {2e} leq x leq 0 $.



              In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.






              share|cite|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$

                This is not a full answer.



                I had to work a similar problem years ago and I used two different series expansions.



                Built at $x=-frac 1e$
                $$W(x)=-1+p-frac{p^2}{3}+frac{11 p^3}{72}-frac{43 p^4}{540}+frac{769
                p^5}{17280}-frac{221 p^6}{8505}+frac{680863 p^7}{43545600}-frac{1963
                p^8}{204120}+frac{226287557 p^9}{37623398400}-frac{5776369
                p^{10}}{1515591000}+frac{169709463197 p^{11}}{69528040243200}-frac{1118511313
                p^{12}}{709296588000}+Oleft(p^{13}right)$$                 in which $p= sqrt{(2(1+e x)}$



                Built at $x=0$
                $$W(x)=x-x^2+frac{3 x^3}{2}-frac{8 x^4}{3}+frac{125 x^5}{24}-frac{54
                x^6}{5}+frac{16807 x^7}{720}-frac{16384 x^8}{315}+frac{531441
                x^9}{4480}-frac{156250 x^{10}}{567}+frac{2357947691
                x^{11}}{3628800}-frac{2985984 x^{12}}{1925}+Oleft(x^{13}right)$$                 For sure, you can truncate or extend these series.



                Numerical calculations show that the first one would be used for $-frac1 {e} leq x leq -frac1 {2e} $ and the second one for $-frac1 {2e} leq x leq 0 $.



                In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.






                share|cite|improve this answer









                $endgroup$



                This is not a full answer.



                I had to work a similar problem years ago and I used two different series expansions.



                Built at $x=-frac 1e$
                $$W(x)=-1+p-frac{p^2}{3}+frac{11 p^3}{72}-frac{43 p^4}{540}+frac{769
                p^5}{17280}-frac{221 p^6}{8505}+frac{680863 p^7}{43545600}-frac{1963
                p^8}{204120}+frac{226287557 p^9}{37623398400}-frac{5776369
                p^{10}}{1515591000}+frac{169709463197 p^{11}}{69528040243200}-frac{1118511313
                p^{12}}{709296588000}+Oleft(p^{13}right)$$                 in which $p= sqrt{(2(1+e x)}$



                Built at $x=0$
                $$W(x)=x-x^2+frac{3 x^3}{2}-frac{8 x^4}{3}+frac{125 x^5}{24}-frac{54
                x^6}{5}+frac{16807 x^7}{720}-frac{16384 x^8}{315}+frac{531441
                x^9}{4480}-frac{156250 x^{10}}{567}+frac{2357947691
                x^{11}}{3628800}-frac{2985984 x^{12}}{1925}+Oleft(x^{13}right)$$                 For sure, you can truncate or extend these series.



                Numerical calculations show that the first one would be used for $-frac1 {e} leq x leq -frac1 {2e} $ and the second one for $-frac1 {2e} leq x leq 0 $.



                In practice, I transformed (for the same accuracy) these expansions into Padé approximants but it would not be of any use if you need to perform integrations.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 30 at 11:17









                Claude LeiboviciClaude Leibovici

                125k1158135




                125k1158135






























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