Notation for repeated composition of functions












5












$begingroup$


I have a repeated composition of functions ${T_n}(z) = {tau _0} circ {tau _1} circ {tau _2} circ cdots circ {tau _n}(z)$



By analogy with $sumlimits_{i = 1}^n {} ,prodlimits_{i = 1}^n {} ,bigcuplimits_{i = 1}^n {} ,bigcaplimits_{i = 1}^n {} ,$
I want to write ${T_n}(z) = left( {mathop circ limits_{i = 0}^n {tau _i}} right)(z)$ or even ${T_n}(z) = {mathop circ limits_{i = 0}^n {tau _i}} (z)$. Can I do this?






notation function-and-relation-composition







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$endgroup$

















    5












    $begingroup$


    I have a repeated composition of functions ${T_n}(z) = {tau _0} circ {tau _1} circ {tau _2} circ cdots circ {tau _n}(z)$



    By analogy with $sumlimits_{i = 1}^n {} ,prodlimits_{i = 1}^n {} ,bigcuplimits_{i = 1}^n {} ,bigcaplimits_{i = 1}^n {} ,$
    I want to write ${T_n}(z) = left( {mathop circ limits_{i = 0}^n {tau _i}} right)(z)$ or even ${T_n}(z) = {mathop circ limits_{i = 0}^n {tau _i}} (z)$. Can I do this?






    notation function-and-relation-composition







    share|cite|improve this question









    $endgroup$















      5












      5








      5


      1



      $begingroup$


      I have a repeated composition of functions ${T_n}(z) = {tau _0} circ {tau _1} circ {tau _2} circ cdots circ {tau _n}(z)$



      By analogy with $sumlimits_{i = 1}^n {} ,prodlimits_{i = 1}^n {} ,bigcuplimits_{i = 1}^n {} ,bigcaplimits_{i = 1}^n {} ,$
      I want to write ${T_n}(z) = left( {mathop circ limits_{i = 0}^n {tau _i}} right)(z)$ or even ${T_n}(z) = {mathop circ limits_{i = 0}^n {tau _i}} (z)$. Can I do this?






      notation function-and-relation-composition







      share|cite|improve this question









      $endgroup$




      I have a repeated composition of functions ${T_n}(z) = {tau _0} circ {tau _1} circ {tau _2} circ cdots circ {tau _n}(z)$



      By analogy with $sumlimits_{i = 1}^n {} ,prodlimits_{i = 1}^n {} ,bigcuplimits_{i = 1}^n {} ,bigcaplimits_{i = 1}^n {} ,$
      I want to write ${T_n}(z) = left( {mathop circ limits_{i = 0}^n {tau _i}} right)(z)$ or even ${T_n}(z) = {mathop circ limits_{i = 0}^n {tau _i}} (z)$. Can I do this?






      notation function-and-relation-composition




      notation function-and-relation-composition






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      share|cite|improve this question











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      share|cite|improve this question










      asked Sep 10 '14 at 13:07









      BudgieJaneBudgieJane

      667




      667






















          3 Answers
          3






          active

          oldest

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          3












          $begingroup$

          Let $f_n(x)$ be a sequence of functions indexed by $n$.



          Define a new sequence of functions $F_k(x)$ indexed by $k$:



          $$
          F_k(x) =
          begin{cases}
          f_0(x) &: k=0\
          (f_kcirc F_{k-1})(x)&: kgt 0
          end{cases}
          $$



          So you have, for example
          $$
          begin{align}
          F_0(x) &= f_0(x)\
          F_1(x) &= (f_1 circ f_0)(x)\
          F_2(x) &= (f_2 circ F_1)(x) = (f_2 circ f_1 circ f_0)(x)\
          &dots\
          F_{n}(x) &= (f_ncirc f_{n-1} circ cdots circ f_0)(x)\
          end{align}
          $$



          Then for your composition $f_ncirc f_{n-1} circ cdots circ f_0$ of $n$ terms you can simply write $F_n$.



          Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 circ f_1 circ f_2$.






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do:
            $$
            overset{n}{underset{k=0}bigcirc} f_k quad text{ or } quad overset{0}{underset{k=n}bigcirc} f_k .
            $$
            Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $displaystylebigcap$ in $displaystylebigcap_{k=0}^n A_k$ or to $displaystylebigoplus$. Before the begin{document} I'd put newcommand{Circ}{blah blah blah} (with a capital "C" distinguishing it from circ.






            share|cite|improve this answer











            $endgroup$





















              2












              $begingroup$

              Basically you can do anything. Notice, however, that the classical operators $sum$ and $prod$ do not coincide with the symbols they represent. And old books used to have $sum$ instead of $bigcup$ and $prod$ instead of $bigcap$.



              I personally understand the notation $mathop{circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$mathop{rm C}limits_{n=1}^N $$ or $$mathop{rm K}limits_{n=1}^N ?$$
              Anyway, I don't like them either...






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                $endgroup$
                – BudgieJane
                Sep 10 '14 at 17:55











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              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              3












              $begingroup$

              Let $f_n(x)$ be a sequence of functions indexed by $n$.



              Define a new sequence of functions $F_k(x)$ indexed by $k$:



              $$
              F_k(x) =
              begin{cases}
              f_0(x) &: k=0\
              (f_kcirc F_{k-1})(x)&: kgt 0
              end{cases}
              $$



              So you have, for example
              $$
              begin{align}
              F_0(x) &= f_0(x)\
              F_1(x) &= (f_1 circ f_0)(x)\
              F_2(x) &= (f_2 circ F_1)(x) = (f_2 circ f_1 circ f_0)(x)\
              &dots\
              F_{n}(x) &= (f_ncirc f_{n-1} circ cdots circ f_0)(x)\
              end{align}
              $$



              Then for your composition $f_ncirc f_{n-1} circ cdots circ f_0$ of $n$ terms you can simply write $F_n$.



              Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 circ f_1 circ f_2$.






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                Let $f_n(x)$ be a sequence of functions indexed by $n$.



                Define a new sequence of functions $F_k(x)$ indexed by $k$:



                $$
                F_k(x) =
                begin{cases}
                f_0(x) &: k=0\
                (f_kcirc F_{k-1})(x)&: kgt 0
                end{cases}
                $$



                So you have, for example
                $$
                begin{align}
                F_0(x) &= f_0(x)\
                F_1(x) &= (f_1 circ f_0)(x)\
                F_2(x) &= (f_2 circ F_1)(x) = (f_2 circ f_1 circ f_0)(x)\
                &dots\
                F_{n}(x) &= (f_ncirc f_{n-1} circ cdots circ f_0)(x)\
                end{align}
                $$



                Then for your composition $f_ncirc f_{n-1} circ cdots circ f_0$ of $n$ terms you can simply write $F_n$.



                Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 circ f_1 circ f_2$.






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Let $f_n(x)$ be a sequence of functions indexed by $n$.



                  Define a new sequence of functions $F_k(x)$ indexed by $k$:



                  $$
                  F_k(x) =
                  begin{cases}
                  f_0(x) &: k=0\
                  (f_kcirc F_{k-1})(x)&: kgt 0
                  end{cases}
                  $$



                  So you have, for example
                  $$
                  begin{align}
                  F_0(x) &= f_0(x)\
                  F_1(x) &= (f_1 circ f_0)(x)\
                  F_2(x) &= (f_2 circ F_1)(x) = (f_2 circ f_1 circ f_0)(x)\
                  &dots\
                  F_{n}(x) &= (f_ncirc f_{n-1} circ cdots circ f_0)(x)\
                  end{align}
                  $$



                  Then for your composition $f_ncirc f_{n-1} circ cdots circ f_0$ of $n$ terms you can simply write $F_n$.



                  Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 circ f_1 circ f_2$.






                  share|cite|improve this answer









                  $endgroup$



                  Let $f_n(x)$ be a sequence of functions indexed by $n$.



                  Define a new sequence of functions $F_k(x)$ indexed by $k$:



                  $$
                  F_k(x) =
                  begin{cases}
                  f_0(x) &: k=0\
                  (f_kcirc F_{k-1})(x)&: kgt 0
                  end{cases}
                  $$



                  So you have, for example
                  $$
                  begin{align}
                  F_0(x) &= f_0(x)\
                  F_1(x) &= (f_1 circ f_0)(x)\
                  F_2(x) &= (f_2 circ F_1)(x) = (f_2 circ f_1 circ f_0)(x)\
                  &dots\
                  F_{n}(x) &= (f_ncirc f_{n-1} circ cdots circ f_0)(x)\
                  end{align}
                  $$



                  Then for your composition $f_ncirc f_{n-1} circ cdots circ f_0$ of $n$ terms you can simply write $F_n$.



                  Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 circ f_1 circ f_2$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Oct 24 '16 at 1:36









                  chharveychharvey

                  1,37731839




                  1,37731839























                      3












                      $begingroup$

                      If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do:
                      $$
                      overset{n}{underset{k=0}bigcirc} f_k quad text{ or } quad overset{0}{underset{k=n}bigcirc} f_k .
                      $$
                      Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $displaystylebigcap$ in $displaystylebigcap_{k=0}^n A_k$ or to $displaystylebigoplus$. Before the begin{document} I'd put newcommand{Circ}{blah blah blah} (with a capital "C" distinguishing it from circ.






                      share|cite|improve this answer











                      $endgroup$


















                        3












                        $begingroup$

                        If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do:
                        $$
                        overset{n}{underset{k=0}bigcirc} f_k quad text{ or } quad overset{0}{underset{k=n}bigcirc} f_k .
                        $$
                        Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $displaystylebigcap$ in $displaystylebigcap_{k=0}^n A_k$ or to $displaystylebigoplus$. Before the begin{document} I'd put newcommand{Circ}{blah blah blah} (with a capital "C" distinguishing it from circ.






                        share|cite|improve this answer











                        $endgroup$
















                          3












                          3








                          3





                          $begingroup$

                          If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do:
                          $$
                          overset{n}{underset{k=0}bigcirc} f_k quad text{ or } quad overset{0}{underset{k=n}bigcirc} f_k .
                          $$
                          Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $displaystylebigcap$ in $displaystylebigcap_{k=0}^n A_k$ or to $displaystylebigoplus$. Before the begin{document} I'd put newcommand{Circ}{blah blah blah} (with a capital "C" distinguishing it from circ.






                          share|cite|improve this answer











                          $endgroup$



                          If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do:
                          $$
                          overset{n}{underset{k=0}bigcirc} f_k quad text{ or } quad overset{0}{underset{k=n}bigcirc} f_k .
                          $$
                          Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $displaystylebigcap$ in $displaystylebigcap_{k=0}^n A_k$ or to $displaystylebigoplus$. Before the begin{document} I'd put newcommand{Circ}{blah blah blah} (with a capital "C" distinguishing it from circ.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Dec 21 '18 at 10:41









                          Sik Feng Cheong

                          17710




                          17710










                          answered Jan 8 '15 at 23:28









                          Michael HardyMichael Hardy

                          1




                          1























                              2












                              $begingroup$

                              Basically you can do anything. Notice, however, that the classical operators $sum$ and $prod$ do not coincide with the symbols they represent. And old books used to have $sum$ instead of $bigcup$ and $prod$ instead of $bigcap$.



                              I personally understand the notation $mathop{circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$mathop{rm C}limits_{n=1}^N $$ or $$mathop{rm K}limits_{n=1}^N ?$$
                              Anyway, I don't like them either...






                              share|cite|improve this answer









                              $endgroup$













                              • $begingroup$
                                I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                                $endgroup$
                                – BudgieJane
                                Sep 10 '14 at 17:55
















                              2












                              $begingroup$

                              Basically you can do anything. Notice, however, that the classical operators $sum$ and $prod$ do not coincide with the symbols they represent. And old books used to have $sum$ instead of $bigcup$ and $prod$ instead of $bigcap$.



                              I personally understand the notation $mathop{circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$mathop{rm C}limits_{n=1}^N $$ or $$mathop{rm K}limits_{n=1}^N ?$$
                              Anyway, I don't like them either...






                              share|cite|improve this answer









                              $endgroup$













                              • $begingroup$
                                I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                                $endgroup$
                                – BudgieJane
                                Sep 10 '14 at 17:55














                              2












                              2








                              2





                              $begingroup$

                              Basically you can do anything. Notice, however, that the classical operators $sum$ and $prod$ do not coincide with the symbols they represent. And old books used to have $sum$ instead of $bigcup$ and $prod$ instead of $bigcap$.



                              I personally understand the notation $mathop{circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$mathop{rm C}limits_{n=1}^N $$ or $$mathop{rm K}limits_{n=1}^N ?$$
                              Anyway, I don't like them either...






                              share|cite|improve this answer









                              $endgroup$



                              Basically you can do anything. Notice, however, that the classical operators $sum$ and $prod$ do not coincide with the symbols they represent. And old books used to have $sum$ instead of $bigcup$ and $prod$ instead of $bigcap$.



                              I personally understand the notation $mathop{circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$mathop{rm C}limits_{n=1}^N $$ or $$mathop{rm K}limits_{n=1}^N ?$$
                              Anyway, I don't like them either...







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Sep 10 '14 at 13:14









                              SiminoreSiminore

                              30.5k33569




                              30.5k33569












                              • $begingroup$
                                I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                                $endgroup$
                                – BudgieJane
                                Sep 10 '14 at 17:55


















                              • $begingroup$
                                I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                                $endgroup$
                                – BudgieJane
                                Sep 10 '14 at 17:55
















                              $begingroup$
                              I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                              $endgroup$
                              – BudgieJane
                              Sep 10 '14 at 17:55




                              $begingroup$
                              I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction.
                              $endgroup$
                              – BudgieJane
                              Sep 10 '14 at 17:55


















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