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












4












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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?






calculus integration proof-theory







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  • 1




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    5 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?






calculus integration proof-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Differential algebra is a galois like approach to proving such things. pdfs.semanticscholar.org/3d42/…
    $endgroup$
    – Charlie Frohman
    5 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?






calculus integration proof-theory







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




calculus integration proof-theory






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share|cite|improve this question













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edited 5 hours ago









Bernard

123k741117




123k741117










asked 5 hours ago









Rithik KapoorRithik Kapoor

3029




3029








  • 1




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














  • 1




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








1




1




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




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










2 Answers
2






active

oldest

votes


















9












$begingroup$

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
    5 hours 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









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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    9












    $begingroup$

    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
      5 hours ago
















    9












    $begingroup$

    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
      5 hours ago














    9












    9








    9





    $begingroup$

    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 4 hours ago

























    answered 5 hours ago









    Robert IsraelRobert Israel

    329k23217470




    329k23217470












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


















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
















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




    $begingroup$
    Okay then how can we prove the derivative of an elementary function is always an elementary function?
    $endgroup$
    – Rithik Kapoor
    5 hours 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 5 hours ago









        El EctricEl Ectric

        14910




        14910






























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