How can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of...












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We know that the derivative of some non-elementary functions can be expressed in elementary functions. For example $ frac{d}{dx} Si(x)= frac{sin(x)}{x} $



So similarly are there any non-elementary functions whose integrals can be expressed in elementary functions?



If not then how can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?










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  • 2




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    8 hours ago
















4












$begingroup$


We know that the derivative of some non-elementary functions can be expressed in elementary functions. For example $ frac{d}{dx} Si(x)= frac{sin(x)}{x} $



So similarly are there any non-elementary functions whose integrals can be expressed in elementary functions?



If not then how can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    8 hours ago














4












4








4


1



$begingroup$


We know that the derivative of some non-elementary functions can be expressed in elementary functions. For example $ frac{d}{dx} Si(x)= frac{sin(x)}{x} $



So similarly are there any non-elementary functions whose integrals can be expressed in elementary functions?



If not then how can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?










share|cite|improve this question











$endgroup$




We know that the derivative of some non-elementary functions can be expressed in elementary functions. For example $ frac{d}{dx} Si(x)= frac{sin(x)}{x} $



So similarly are there any non-elementary functions whose integrals can be expressed in elementary functions?



If not then how can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?







calculus integration proof-theory






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edited 8 hours ago









Bernard

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123k741117










asked 8 hours ago









Rithik KapoorRithik Kapoor

30210




30210








  • 2




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    8 hours ago














  • 2




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    8 hours ago








2




2




$begingroup$
Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
$endgroup$
– Charlie Frohman
8 hours ago




$begingroup$
Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
$endgroup$
– Charlie Frohman
8 hours ago










3 Answers
3






active

oldest

votes


















10












$begingroup$

The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.



EDIT: More formally, by definition an elementary function is obtained from
complex constants and the variable $x$ by a finite number of steps of the following forms:




  1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.

  2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.

  3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).


To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
the result is true for elementary functions obtained in at most $n$ steps.
If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.






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$endgroup$













  • $begingroup$
    Okay then how can we prove the derivative of an elementary function is always an elementary function?
    $endgroup$
    – Rithik Kapoor
    8 hours ago










  • $begingroup$
    @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
    $endgroup$
    – The Great Duck
    2 hours ago










  • $begingroup$
    Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
    $endgroup$
    – The Great Duck
    2 hours ago










  • $begingroup$
    @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
    $endgroup$
    – J. M. is not a mathematician
    1 min ago



















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$begingroup$

No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.



An anti-derivative of a non-elementary function cannot be an elementary function.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Yes, and I can provide a simple counter-example.



    Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.



    This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.



    For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.



    However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.



    In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Why is a piecewise function with elementary cases not elementary?
      $endgroup$
      – J. M. is not a mathematician
      1 hour ago










    • $begingroup$
      @J.M.isnotamathematician an infinite number of cases?
      $endgroup$
      – The Great Duck
      1 hour ago










    • $begingroup$
      I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
      $endgroup$
      – J. M. is not a mathematician
      1 hour ago










    • $begingroup$
      @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
      $endgroup$
      – The Great Duck
      1 hour ago






    • 1




      $begingroup$
      "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
      $endgroup$
      – J. M. is not a mathematician
      52 mins ago











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    3 Answers
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    3 Answers
    3






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    active

    oldest

    votes






    active

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    10












    $begingroup$

    The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.



    EDIT: More formally, by definition an elementary function is obtained from
    complex constants and the variable $x$ by a finite number of steps of the following forms:




    1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.

    2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.

    3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).


    To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
    the result is true for elementary functions obtained in at most $n$ steps.
    If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Okay then how can we prove the derivative of an elementary function is always an elementary function?
      $endgroup$
      – Rithik Kapoor
      8 hours ago










    • $begingroup$
      @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
      $endgroup$
      – J. M. is not a mathematician
      1 min ago
















    10












    $begingroup$

    The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.



    EDIT: More formally, by definition an elementary function is obtained from
    complex constants and the variable $x$ by a finite number of steps of the following forms:




    1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.

    2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.

    3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).


    To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
    the result is true for elementary functions obtained in at most $n$ steps.
    If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Okay then how can we prove the derivative of an elementary function is always an elementary function?
      $endgroup$
      – Rithik Kapoor
      8 hours ago










    • $begingroup$
      @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
      $endgroup$
      – J. M. is not a mathematician
      1 min ago














    10












    10








    10





    $begingroup$

    The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.



    EDIT: More formally, by definition an elementary function is obtained from
    complex constants and the variable $x$ by a finite number of steps of the following forms:




    1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.

    2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.

    3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).


    To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
    the result is true for elementary functions obtained in at most $n$ steps.
    If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.






    share|cite|improve this answer











    $endgroup$



    The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.



    EDIT: More formally, by definition an elementary function is obtained from
    complex constants and the variable $x$ by a finite number of steps of the following forms:




    1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.

    2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.

    3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).


    To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
    the result is true for elementary functions obtained in at most $n$ steps.
    If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 8 hours ago

























    answered 8 hours ago









    Robert IsraelRobert Israel

    329k23218472




    329k23218472












    • $begingroup$
      Okay then how can we prove the derivative of an elementary function is always an elementary function?
      $endgroup$
      – Rithik Kapoor
      8 hours ago










    • $begingroup$
      @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
      $endgroup$
      – J. M. is not a mathematician
      1 min ago


















    • $begingroup$
      Okay then how can we prove the derivative of an elementary function is always an elementary function?
      $endgroup$
      – Rithik Kapoor
      8 hours ago










    • $begingroup$
      @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
      $endgroup$
      – The Great Duck
      2 hours ago










    • $begingroup$
      @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
      $endgroup$
      – J. M. is not a mathematician
      1 min ago
















    $begingroup$
    Okay then how can we prove the derivative of an elementary function is always an elementary function?
    $endgroup$
    – Rithik Kapoor
    8 hours ago




    $begingroup$
    Okay then how can we prove the derivative of an elementary function is always an elementary function?
    $endgroup$
    – Rithik Kapoor
    8 hours ago












    $begingroup$
    @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
    $endgroup$
    – The Great Duck
    2 hours ago




    $begingroup$
    @RithikKapoor Frankly sir, common sense. Writing an explicit formal proof might be tricky but we already know that any composition of elementary functions can be handled with the chain, product, division, and addition rules. If you're asking how to formalize it, fair enough. However if you are asking "How do I know this is true" then it follows by simple observance.
    $endgroup$
    – The Great Duck
    2 hours ago












    $begingroup$
    Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
    $endgroup$
    – The Great Duck
    2 hours ago




    $begingroup$
    Query. What if we include the Heaviside step function as an elementary function? Would this have any affect on the answer?
    $endgroup$
    – The Great Duck
    2 hours ago












    $begingroup$
    @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
    $endgroup$
    – J. M. is not a mathematician
    1 min ago




    $begingroup$
    @TheGreatDuck, before anything else, what would you say about the function $frac{x+sqrt{x^2}}{2x}$?
    $endgroup$
    – J. M. is not a mathematician
    1 min ago











    1












    $begingroup$

    No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.



    An anti-derivative of a non-elementary function cannot be an elementary function.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.



      An anti-derivative of a non-elementary function cannot be an elementary function.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.



        An anti-derivative of a non-elementary function cannot be an elementary function.






        share|cite|improve this answer









        $endgroup$



        No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.



        An anti-derivative of a non-elementary function cannot be an elementary function.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 8 hours ago









        El EctricEl Ectric

        14910




        14910























            0












            $begingroup$

            Yes, and I can provide a simple counter-example.



            Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.



            This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.



            For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.



            However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.



            In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Why is a piecewise function with elementary cases not elementary?
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician an infinite number of cases?
              $endgroup$
              – The Great Duck
              1 hour ago










            • $begingroup$
              I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
              $endgroup$
              – The Great Duck
              1 hour ago






            • 1




              $begingroup$
              "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
              $endgroup$
              – J. M. is not a mathematician
              52 mins ago
















            0












            $begingroup$

            Yes, and I can provide a simple counter-example.



            Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.



            This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.



            For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.



            However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.



            In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Why is a piecewise function with elementary cases not elementary?
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician an infinite number of cases?
              $endgroup$
              – The Great Duck
              1 hour ago










            • $begingroup$
              I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
              $endgroup$
              – The Great Duck
              1 hour ago






            • 1




              $begingroup$
              "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
              $endgroup$
              – J. M. is not a mathematician
              52 mins ago














            0












            0








            0





            $begingroup$

            Yes, and I can provide a simple counter-example.



            Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.



            This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.



            For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.



            However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.



            In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.






            share|cite|improve this answer









            $endgroup$



            Yes, and I can provide a simple counter-example.



            Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.



            This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.



            For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.



            However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.



            In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 hours ago









            The Great DuckThe Great Duck

            24932047




            24932047












            • $begingroup$
              Why is a piecewise function with elementary cases not elementary?
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician an infinite number of cases?
              $endgroup$
              – The Great Duck
              1 hour ago










            • $begingroup$
              I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
              $endgroup$
              – The Great Duck
              1 hour ago






            • 1




              $begingroup$
              "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
              $endgroup$
              – J. M. is not a mathematician
              52 mins ago


















            • $begingroup$
              Why is a piecewise function with elementary cases not elementary?
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician an infinite number of cases?
              $endgroup$
              – The Great Duck
              1 hour ago










            • $begingroup$
              I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
              $endgroup$
              – J. M. is not a mathematician
              1 hour ago










            • $begingroup$
              @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
              $endgroup$
              – The Great Duck
              1 hour ago






            • 1




              $begingroup$
              "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
              $endgroup$
              – J. M. is not a mathematician
              52 mins ago
















            $begingroup$
            Why is a piecewise function with elementary cases not elementary?
            $endgroup$
            – J. M. is not a mathematician
            1 hour ago




            $begingroup$
            Why is a piecewise function with elementary cases not elementary?
            $endgroup$
            – J. M. is not a mathematician
            1 hour ago












            $begingroup$
            @J.M.isnotamathematician an infinite number of cases?
            $endgroup$
            – The Great Duck
            1 hour ago




            $begingroup$
            @J.M.isnotamathematician an infinite number of cases?
            $endgroup$
            – The Great Duck
            1 hour ago












            $begingroup$
            I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
            $endgroup$
            – J. M. is not a mathematician
            1 hour ago




            $begingroup$
            I am talking about your second sentence. You give a quadratic function with a hole and say that it is nonelementary.
            $endgroup$
            – J. M. is not a mathematician
            1 hour ago












            $begingroup$
            @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
            $endgroup$
            – The Great Duck
            1 hour ago




            $begingroup$
            @J.M.isnotamathematician As I said, there are much messier example and I gave one. The definition of elementary is tenuous at best. Provide a detailed analytical definition and I'll say whether something fits inside it. Until then, there's no real way to tell for sure. Elementary has imo always been a subjective concept. Regardless, I can easily keep cranking up the complexity on the counter-example so it doesn't change the result if I mis-identify some simpler function as being non-elementary.
            $endgroup$
            – The Great Duck
            1 hour ago




            1




            1




            $begingroup$
            "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
            $endgroup$
            – J. M. is not a mathematician
            52 mins ago




            $begingroup$
            "Elementary has imo always been a subjective concept." - in this regard at least, we are in agreement.
            $endgroup$
            – J. M. is not a mathematician
            52 mins ago


















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