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












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












    $begingroup$


    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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 30 at 9:53









      JohannaJohanna

      4,74541640




      4,74541640






















          1 Answer
          1






          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









          $endgroup$














            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3093316%2fsharp-bounds-for-the-principal-branch-of-the-lambert-w-function%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            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






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3093316%2fsharp-bounds-for-the-principal-branch-of-the-lambert-w-function%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

                    Image
                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 This is a simple project showing cubes on Unity, exported to android project and imported on Android studio. On maifest file it showed following warning: Expecting <uses-feature android:name="android.software.leanback" android:required="false" /> tag I have no information about it. And also on default emptyview android project does not use/mention this feature. I know it is required by TV's and such but why its not added by default? Why is this? And Should I concern about it? android unity3d
                    Read more

                    SQL update select statement

                    Image
                    0 I'm trying to update an entry in my database (postgresql). I'm having a problem with updating the query: INSERT INTO prices (price, product_id) SELECT product_price, commn_id FROM products_temp as prod WHERE NOT EXISTS (SELECT * FROM prices od WHERE prod.commn_id = od.product_id) OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) Query works fine when i delete: OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) So it seems to me that it's looping through this operation. My question is, how can I fix it? sql postgresql
                    Read more

                    WPF add header to Image with URL pettitions [duplicate]

                    Image
                    up vote 0 down vote favorite This question already has an answer here: Image source URI requiring authorization 3 answers Add header to all* HTTP requests on a machine 1 answer first of all sorry for my english. Recently I added some security in the backend of my project, and now I need to send a validation code(apiKey) inside the headers to retrieve the images. I have a problem with ALL images in my project that I had attached in my UI. For example, before I had: <Image Height="90" Source="{Binding Video.Thumbnail}"/> Now I need to change my code to create a CUSTOM petition add the apiKey in the header:
                    Read more