Is there an accepted notation for the nth sum/integral of a function?












2












$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the nth integral of a function of n variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of x_1 from 0 to M, then the sum of x_1 from 0 to M, and so on with a total of of N summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    3 hours ago










  • $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    2 hours ago
















2












$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the nth integral of a function of n variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of x_1 from 0 to M, then the sum of x_1 from 0 to M, and so on with a total of of N summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    3 hours ago










  • $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    2 hours ago














2












2








2





$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the nth integral of a function of n variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of x_1 from 0 to M, then the sum of x_1 from 0 to M, and so on with a total of of N summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$




I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the nth integral of a function of n variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of x_1 from 0 to M, then the sum of x_1 from 0 to M, and so on with a total of of N summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation




integration functional-analysis summation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Peace Blaster

















asked 3 hours ago









Peace BlasterPeace Blaster

387




387












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    3 hours ago










  • $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    2 hours ago


















  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    3 hours ago










  • $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    2 hours ago
















$begingroup$
Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
$endgroup$
– Ethan Bolker
3 hours ago




$begingroup$
Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
$endgroup$
– Ethan Bolker
3 hours ago












$begingroup$
you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
$endgroup$
– Arjang
2 hours ago




$begingroup$
you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
$endgroup$
– Arjang
2 hours ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



Here are two suggestions each using just a single $sum$.



$$
sum_{x in [1, 2, ldots, M]^n} f(x)
$$



$$
sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    2 hours ago



















1












$begingroup$

How about something like :
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






share|cite|improve this answer









$endgroup$













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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

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    4












    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      2 hours ago
















    4












    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      2 hours ago














    4












    4








    4





    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$



    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 hours ago









    Ethan BolkerEthan Bolker

    42.7k549113




    42.7k549113












    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      2 hours ago


















    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      2 hours ago
















    $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    2 hours ago




    $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    2 hours ago











    1












    $begingroup$

    How about something like :
    $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



    Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      How about something like :
      $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



      Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        How about something like :
        $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



        Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






        share|cite|improve this answer









        $endgroup$



        How about something like :
        $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



        Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        ArjangArjang

        5,60162363




        5,60162363






























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