Is there a non-elementary function with an elementary derivative and an elementary inverse?












14












$begingroup$


Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary.



There are a couple of equivalent ways to ask my question:




  1. Is there a function $F(x)$ that is non-elementary, but its derivative $F'(x)$ and inverse $F^{-1}(x)$ are both elementary?


  2. Is there an elementary function $f(x)$ whose integral $F(x)$ is non-elementary, but can be expressed as the inverse of some elementary function? (i.e. $F^{-1}(x)$ is elementary)



Here are some non-examples:



The Lambert W function is the inverse of $x e^x$. It is not elementary, but its derivative is not elementary either:
$$W'(x) = frac{W(x)}{x(1 + W(x))}$$



The "exponential integral" $Ei(x)$ is the integral of $int frac{e^x}{x} dx$, which is non-elementary. Its inverse $Ei^{-1}(x)$ is not elementary either, so this is not what I'm looking for.






integration







share|cite|improve this question











$endgroup$












  • $begingroup$
    Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
    $endgroup$
    – Rahul
    Jul 31 '18 at 12:33












  • $begingroup$
    Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
    $endgroup$
    – IV_
    Jan 19 at 23:34












  • $begingroup$
    No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
    $endgroup$
    – IV_
    Jan 19 at 23:36










  • $begingroup$
    This is an old question, but I think you should ask on MO. This very much seems to be research-level.
    $endgroup$
    – YiFan
    Jan 20 at 0:04
















14












$begingroup$


Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary.



There are a couple of equivalent ways to ask my question:




  1. Is there a function $F(x)$ that is non-elementary, but its derivative $F'(x)$ and inverse $F^{-1}(x)$ are both elementary?


  2. Is there an elementary function $f(x)$ whose integral $F(x)$ is non-elementary, but can be expressed as the inverse of some elementary function? (i.e. $F^{-1}(x)$ is elementary)



Here are some non-examples:



The Lambert W function is the inverse of $x e^x$. It is not elementary, but its derivative is not elementary either:
$$W'(x) = frac{W(x)}{x(1 + W(x))}$$



The "exponential integral" $Ei(x)$ is the integral of $int frac{e^x}{x} dx$, which is non-elementary. Its inverse $Ei^{-1}(x)$ is not elementary either, so this is not what I'm looking for.






integration







share|cite|improve this question











$endgroup$












  • $begingroup$
    Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
    $endgroup$
    – Rahul
    Jul 31 '18 at 12:33












  • $begingroup$
    Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
    $endgroup$
    – IV_
    Jan 19 at 23:34












  • $begingroup$
    No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
    $endgroup$
    – IV_
    Jan 19 at 23:36










  • $begingroup$
    This is an old question, but I think you should ask on MO. This very much seems to be research-level.
    $endgroup$
    – YiFan
    Jan 20 at 0:04














14












14








14


5



$begingroup$


Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary.



There are a couple of equivalent ways to ask my question:




  1. Is there a function $F(x)$ that is non-elementary, but its derivative $F'(x)$ and inverse $F^{-1}(x)$ are both elementary?


  2. Is there an elementary function $f(x)$ whose integral $F(x)$ is non-elementary, but can be expressed as the inverse of some elementary function? (i.e. $F^{-1}(x)$ is elementary)



Here are some non-examples:



The Lambert W function is the inverse of $x e^x$. It is not elementary, but its derivative is not elementary either:
$$W'(x) = frac{W(x)}{x(1 + W(x))}$$



The "exponential integral" $Ei(x)$ is the integral of $int frac{e^x}{x} dx$, which is non-elementary. Its inverse $Ei^{-1}(x)$ is not elementary either, so this is not what I'm looking for.






integration







share|cite|improve this question











$endgroup$




Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary.



There are a couple of equivalent ways to ask my question:




  1. Is there a function $F(x)$ that is non-elementary, but its derivative $F'(x)$ and inverse $F^{-1}(x)$ are both elementary?


  2. Is there an elementary function $f(x)$ whose integral $F(x)$ is non-elementary, but can be expressed as the inverse of some elementary function? (i.e. $F^{-1}(x)$ is elementary)



Here are some non-examples:



The Lambert W function is the inverse of $x e^x$. It is not elementary, but its derivative is not elementary either:
$$W'(x) = frac{W(x)}{x(1 + W(x))}$$



The "exponential integral" $Ei(x)$ is the integral of $int frac{e^x}{x} dx$, which is non-elementary. Its inverse $Ei^{-1}(x)$ is not elementary either, so this is not what I'm looking for.






integration




integration






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jul 31 '18 at 12:19







Paul Castle

















asked Jul 31 '18 at 4:58









Paul CastlePaul Castle

576214




576214












  • $begingroup$
    Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
    $endgroup$
    – Rahul
    Jul 31 '18 at 12:33












  • $begingroup$
    Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
    $endgroup$
    – IV_
    Jan 19 at 23:34












  • $begingroup$
    No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
    $endgroup$
    – IV_
    Jan 19 at 23:36










  • $begingroup$
    This is an old question, but I think you should ask on MO. This very much seems to be research-level.
    $endgroup$
    – YiFan
    Jan 20 at 0:04


















  • $begingroup$
    Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
    $endgroup$
    – Rahul
    Jul 31 '18 at 12:33












  • $begingroup$
    Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
    $endgroup$
    – IV_
    Jan 19 at 23:34












  • $begingroup$
    No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
    $endgroup$
    – IV_
    Jan 19 at 23:36










  • $begingroup$
    This is an old question, but I think you should ask on MO. This very much seems to be research-level.
    $endgroup$
    – YiFan
    Jan 20 at 0:04
















$begingroup$
Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
$endgroup$
– Rahul
Jul 31 '18 at 12:33






$begingroup$
Let $G=F^{-1}$. You want $G$ elementary, $G^{-1}$ non-elementary and $(G^{-1})'$ elementary. But $(G^{-1})'=1/(G'circ G^{-1})$, so $1/(G^{-1})' = G'circ G^{-1}$. Does there exist any nontrivial example of $text{elementary} = text{elementary} circ text{non-elementary}$?
$endgroup$
– Rahul
Jul 31 '18 at 12:33














$begingroup$
Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
$endgroup$
– IV_
Jan 19 at 23:34






$begingroup$
Because all compositions $xmapsto h_1(h_2(...(h_n(x))...))$ of algebraic or elementary functions $h_1,...,h_n$ are elementary invertible (a corollary of the theorem in Ritt 1925), the expression of $F^{−1}$ cannot have this form. $F^{−1}$ must therefore be a generalized composition of elementary and algebraic functions. That means, the expression of $F^{−1}$ contains an expression $A(f_1(x),...,f_n(x))$, wherein $A$ is an algebraic function, and $f_1,...,f_n$ are elementary functions, pairwise algebraically independent of each other. No idea if that can help here.
$endgroup$
– IV_
Jan 19 at 23:34














$begingroup$
No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
$endgroup$
– IV_
Jan 19 at 23:36




$begingroup$
No idea how the inverse of $A(f_1(x),...,f_n(x))$ could be presented in closed form.
$endgroup$
– IV_
Jan 19 at 23:36












$begingroup$
This is an old question, but I think you should ask on MO. This very much seems to be research-level.
$endgroup$
– YiFan
Jan 20 at 0:04




$begingroup$
This is an old question, but I think you should ask on MO. This very much seems to be research-level.
$endgroup$
– YiFan
Jan 20 at 0:04










1 Answer
1






active

oldest

votes


















1












$begingroup$

My answer shows only one of the possibly suitable function classes.



Let $c,c_1,c_2$ be constants. A constant function is an elementary function.



Let $Phi$ denote the inverse of $F$: $Phi=F^{-1}$.



According to the question, we have $Phi$ is elementary, and



$$F(x)=int F'(x)dx+c_1,$$



wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.



$F(x)$ is a non-elementary integral.



The elementary functions are differentiable, and their derivatives are also elementary.



Because $Phi$ is elementary, $Phi(x)=int Phi'(x)dx+c_2$, wherein $Phi'$ is an elementary function.



Assume $F$ is integrable.



Applying the Integral of inverse functions to $int F(x)dx$ gives



$$int F(x)dx=xF(x)-int Phi(F(x))dF(x)+c.$$



$$int F(x)dx=xF(x)-int xdF(x)+c$$



$$int F(x)dx=xF(x)-int xF'(x)dx+c$$



Because $F$ is non-elementary, $int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $int xF'(x)dx$ can be elementary or non-elementary.



Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $int xF'(x)dx$ is non-elementary, $int xF'(x)dx$ must be a non-elementary integral.



See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.



I don't know if such functions $F$ you asked for with a non-elementary integral $int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.



Verification of a found $F$ could be difficult: Take a non-elementary integral $int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $Phi(x)$ can be expressed as an elementary expression.



Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?






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%2f2867673%2fis-there-a-non-elementary-function-with-an-elementary-derivative-and-an-elementa%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









    1












    $begingroup$

    My answer shows only one of the possibly suitable function classes.



    Let $c,c_1,c_2$ be constants. A constant function is an elementary function.



    Let $Phi$ denote the inverse of $F$: $Phi=F^{-1}$.



    According to the question, we have $Phi$ is elementary, and



    $$F(x)=int F'(x)dx+c_1,$$



    wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.



    $F(x)$ is a non-elementary integral.



    The elementary functions are differentiable, and their derivatives are also elementary.



    Because $Phi$ is elementary, $Phi(x)=int Phi'(x)dx+c_2$, wherein $Phi'$ is an elementary function.



    Assume $F$ is integrable.



    Applying the Integral of inverse functions to $int F(x)dx$ gives



    $$int F(x)dx=xF(x)-int Phi(F(x))dF(x)+c.$$



    $$int F(x)dx=xF(x)-int xdF(x)+c$$



    $$int F(x)dx=xF(x)-int xF'(x)dx+c$$



    Because $F$ is non-elementary, $int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $int xF'(x)dx$ can be elementary or non-elementary.



    Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $int xF'(x)dx$ is non-elementary, $int xF'(x)dx$ must be a non-elementary integral.



    See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.



    I don't know if such functions $F$ you asked for with a non-elementary integral $int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.



    Verification of a found $F$ could be difficult: Take a non-elementary integral $int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $Phi(x)$ can be expressed as an elementary expression.



    Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      My answer shows only one of the possibly suitable function classes.



      Let $c,c_1,c_2$ be constants. A constant function is an elementary function.



      Let $Phi$ denote the inverse of $F$: $Phi=F^{-1}$.



      According to the question, we have $Phi$ is elementary, and



      $$F(x)=int F'(x)dx+c_1,$$



      wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.



      $F(x)$ is a non-elementary integral.



      The elementary functions are differentiable, and their derivatives are also elementary.



      Because $Phi$ is elementary, $Phi(x)=int Phi'(x)dx+c_2$, wherein $Phi'$ is an elementary function.



      Assume $F$ is integrable.



      Applying the Integral of inverse functions to $int F(x)dx$ gives



      $$int F(x)dx=xF(x)-int Phi(F(x))dF(x)+c.$$



      $$int F(x)dx=xF(x)-int xdF(x)+c$$



      $$int F(x)dx=xF(x)-int xF'(x)dx+c$$



      Because $F$ is non-elementary, $int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $int xF'(x)dx$ can be elementary or non-elementary.



      Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $int xF'(x)dx$ is non-elementary, $int xF'(x)dx$ must be a non-elementary integral.



      See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.



      I don't know if such functions $F$ you asked for with a non-elementary integral $int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.



      Verification of a found $F$ could be difficult: Take a non-elementary integral $int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $Phi(x)$ can be expressed as an elementary expression.



      Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        My answer shows only one of the possibly suitable function classes.



        Let $c,c_1,c_2$ be constants. A constant function is an elementary function.



        Let $Phi$ denote the inverse of $F$: $Phi=F^{-1}$.



        According to the question, we have $Phi$ is elementary, and



        $$F(x)=int F'(x)dx+c_1,$$



        wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.



        $F(x)$ is a non-elementary integral.



        The elementary functions are differentiable, and their derivatives are also elementary.



        Because $Phi$ is elementary, $Phi(x)=int Phi'(x)dx+c_2$, wherein $Phi'$ is an elementary function.



        Assume $F$ is integrable.



        Applying the Integral of inverse functions to $int F(x)dx$ gives



        $$int F(x)dx=xF(x)-int Phi(F(x))dF(x)+c.$$



        $$int F(x)dx=xF(x)-int xdF(x)+c$$



        $$int F(x)dx=xF(x)-int xF'(x)dx+c$$



        Because $F$ is non-elementary, $int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $int xF'(x)dx$ can be elementary or non-elementary.



        Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $int xF'(x)dx$ is non-elementary, $int xF'(x)dx$ must be a non-elementary integral.



        See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.



        I don't know if such functions $F$ you asked for with a non-elementary integral $int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.



        Verification of a found $F$ could be difficult: Take a non-elementary integral $int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $Phi(x)$ can be expressed as an elementary expression.



        Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?






        share|cite|improve this answer











        $endgroup$



        My answer shows only one of the possibly suitable function classes.



        Let $c,c_1,c_2$ be constants. A constant function is an elementary function.



        Let $Phi$ denote the inverse of $F$: $Phi=F^{-1}$.



        According to the question, we have $Phi$ is elementary, and



        $$F(x)=int F'(x)dx+c_1,$$



        wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.



        $F(x)$ is a non-elementary integral.



        The elementary functions are differentiable, and their derivatives are also elementary.



        Because $Phi$ is elementary, $Phi(x)=int Phi'(x)dx+c_2$, wherein $Phi'$ is an elementary function.



        Assume $F$ is integrable.



        Applying the Integral of inverse functions to $int F(x)dx$ gives



        $$int F(x)dx=xF(x)-int Phi(F(x))dF(x)+c.$$



        $$int F(x)dx=xF(x)-int xdF(x)+c$$



        $$int F(x)dx=xF(x)-int xF'(x)dx+c$$



        Because $F$ is non-elementary, $int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $int xF'(x)dx$ can be elementary or non-elementary.



        Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $int xF'(x)dx$ is non-elementary, $int xF'(x)dx$ must be a non-elementary integral.



        See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.



        I don't know if such functions $F$ you asked for with a non-elementary integral $int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.



        Verification of a found $F$ could be difficult: Take a non-elementary integral $int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $Phi(x)$ can be expressed as an elementary expression.



        Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 27 at 17:56

























        answered Jan 19 at 18:31









        IV_IV_

        1,345525




        1,345525






























            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%2f2867673%2fis-there-a-non-elementary-function-with-an-elementary-derivative-and-an-elementa%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

            Mario Kart Wii

            Image
            Please consider supporting The Cutting Room Floor on Patreon. Thanks for all your support! Mario Kart Wii From The Cutting Room Floor Jump to: navigation, search Mario Kart Wii Developer : Nintendo Publisher : Nintendo Platform : Wii Released in JP : April 10, 2008 Released in US : April 27, 2008 Released in EU : April 11, 2008 Released in AU : April 24, 2008 Released in KR : April 30, 2009 This game has unused animations. This game has unused areas. This game has unused code. This game has unused objects. This game has unused modes / minigames. This game has unused graphics. This game has unused models. This game has unused items. This game has unused music. This game has regional differences. This game has a notes page This game has a prerelease article Mario Kart Wii is the sixth game in the series (not counting Arcade editions), which intro...
            Read more

            The Binding of Isaac: Rebirth/Afterbirth

            Image
            Please consider supporting The Cutting Room Floor on Patreon. Thanks for all your support! The Binding of Isaac: Rebirth/Afterbirth From The Cutting Room Floor < The Binding of Isaac: Rebirth Jump to: navigation, search This is a sub-page of The Binding of Isaac: Rebirth . Afterbirth is the first expansion pack for Rebirth and features a variety of improvements. Contents 1 Unused Graphics 1.1 Old Man Keeper 1.2 Skinless Hush 1.3 Miscellaneous Unused Hush Sprites 1.4 Generosity achievement 1.5 Afterbirth+ Room Editor 1.6 Bearded Ultra Greed 1.7 Early Ultra Greed 1.8 Unused Brownie Expression 1.9 Frail's Spinning Head 1.10 Mom's Dead Hand Alt 2 Unused Audio 2.1 Unused Dead Womb Layer 2.2 Unused Hush Battle Outro 2.3 The Stain Grunt Unused Graphics Old Man Keeper Image file: gfxuimain me...
            Read more

            What does “Dominus providebit” mean?

            Image
            5 Does "Dominus providebit" mean "The Lord will provide" or "The Lord provides"? I once had a bit of an argument with a guy who studied Latin over that. It's an inscription that appears on the rim of some old Swiss coins that I have. meaning latin-to-english-translation grammar-identification share | improve this question edited Jan 18 at 15:27 V2Blast 105 4 asked Jan 17 at 22:49 Donald K. Donald K. 26 1 ...
            Read more