Fractional/Integer Based integrals












2












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I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?






integration definite-integrals soft-question indefinite-integrals big-list







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  • $begingroup$
    Measure theory? Real analysis?
    $endgroup$
    – clathratus
    Jan 2 at 4:55






  • 1




    $begingroup$
    On which page...?
    $endgroup$
    – Darkrai
    Jan 2 at 10:51










  • $begingroup$
    @Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
    $endgroup$
    – user150203
    Jan 6 at 12:03










  • $begingroup$
    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
    $endgroup$
    – user647486
    Mar 22 at 11:46










  • $begingroup$
    Post it as an answer and I'll award you the points.
    $endgroup$
    – user150203
    Mar 22 at 11:47
















2












$begingroup$


I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?






integration definite-integrals soft-question indefinite-integrals big-list







share|cite|improve this question









$endgroup$





This question had a bounty worth +500
reputation from Community♦ that ended 14 hours ago. Grace period ends in 9 hours


Looking for an answer drawing from credible and/or official sources.
















  • $begingroup$
    Measure theory? Real analysis?
    $endgroup$
    – clathratus
    Jan 2 at 4:55






  • 1




    $begingroup$
    On which page...?
    $endgroup$
    – Darkrai
    Jan 2 at 10:51










  • $begingroup$
    @Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
    $endgroup$
    – user150203
    Jan 6 at 12:03










  • $begingroup$
    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
    $endgroup$
    – user647486
    Mar 22 at 11:46










  • $begingroup$
    Post it as an answer and I'll award you the points.
    $endgroup$
    – user150203
    Mar 22 at 11:47














2












2








2





$begingroup$


I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?






integration definite-integrals soft-question indefinite-integrals big-list







share|cite|improve this question









$endgroup$




I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?






integration definite-integrals soft-question indefinite-integrals big-list




integration definite-integrals soft-question indefinite-integrals big-list






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 2 at 3:31







user150203













This question had a bounty worth +500
reputation from Community♦ that ended 14 hours ago. Grace period ends in 9 hours


Looking for an answer drawing from credible and/or official sources.








This question had a bounty worth +500
reputation from Community♦ that ended 14 hours ago. Grace period ends in 9 hours


Looking for an answer drawing from credible and/or official sources.














  • $begingroup$
    Measure theory? Real analysis?
    $endgroup$
    – clathratus
    Jan 2 at 4:55






  • 1




    $begingroup$
    On which page...?
    $endgroup$
    – Darkrai
    Jan 2 at 10:51










  • $begingroup$
    @Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
    $endgroup$
    – user150203
    Jan 6 at 12:03










  • $begingroup$
    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
    $endgroup$
    – user647486
    Mar 22 at 11:46










  • $begingroup$
    Post it as an answer and I'll award you the points.
    $endgroup$
    – user150203
    Mar 22 at 11:47


















  • $begingroup$
    Measure theory? Real analysis?
    $endgroup$
    – clathratus
    Jan 2 at 4:55






  • 1




    $begingroup$
    On which page...?
    $endgroup$
    – Darkrai
    Jan 2 at 10:51










  • $begingroup$
    @Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
    $endgroup$
    – user150203
    Jan 6 at 12:03










  • $begingroup$
    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
    $endgroup$
    – user647486
    Mar 22 at 11:46










  • $begingroup$
    Post it as an answer and I'll award you the points.
    $endgroup$
    – user150203
    Mar 22 at 11:47
















$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55




$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55




1




1




$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51




$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51












$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03




$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03












$begingroup$
$y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46




$begingroup$
$y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46












$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47




$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47










2 Answers
2






active

oldest

votes


















2












$begingroup$

$y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.



In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.



They might appear when working with the Gauss transformation $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    It's forcing me to wait 24 hours. Will award tomorrow this time.
    $endgroup$
    – user150203
    Mar 22 at 11:50



















0












$begingroup$

Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.






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









    2












    $begingroup$

    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.



    In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.



    They might appear when working with the Gauss transformation $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      It's forcing me to wait 24 hours. Will award tomorrow this time.
      $endgroup$
      – user150203
      Mar 22 at 11:50
















    2












    $begingroup$

    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.



    In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.



    They might appear when working with the Gauss transformation $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      It's forcing me to wait 24 hours. Will award tomorrow this time.
      $endgroup$
      – user150203
      Mar 22 at 11:50














    2












    2








    2





    $begingroup$

    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.



    In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.



    They might appear when working with the Gauss transformation $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.






    share|cite|improve this answer











    $endgroup$



    $y={x}$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.



    In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.



    They might appear when working with the Gauss transformation $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Mar 22 at 12:04

























    answered Mar 22 at 11:49









    user647486user647486

    582110




    582110












    • $begingroup$
      It's forcing me to wait 24 hours. Will award tomorrow this time.
      $endgroup$
      – user150203
      Mar 22 at 11:50


















    • $begingroup$
      It's forcing me to wait 24 hours. Will award tomorrow this time.
      $endgroup$
      – user150203
      Mar 22 at 11:50
















    $begingroup$
    It's forcing me to wait 24 hours. Will award tomorrow this time.
    $endgroup$
    – user150203
    Mar 22 at 11:50




    $begingroup$
    It's forcing me to wait 24 hours. Will award tomorrow this time.
    $endgroup$
    – user150203
    Mar 22 at 11:50











    0












    $begingroup$

    Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
    $$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
    Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
      $$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
      Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
        $$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
        Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.






        share|cite|improve this answer









        $endgroup$



        Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
        $$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
        Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        ServaesServaes

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