Possible definitions of exponential function

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I was wondering how many definitions of exponential functions can we think of. The basic ones could be:



$$e^x:=sum_k=0^inftyfracx^kk!$$
also
$$e^x:=lim_ntoinftybigg(1+fracxnbigg)^n$$
or this one:
Define $e^x:mathbbRrightarrowmathbbR\$ as unique function satisfying:
beginalign
e^xgeq x+1\
forall x,yinmathbbR:e^x+y=e^xe^y
endalign
Can anyone come up with something unusual? (Possibly with some explanation or references).






real-analysis definition big-list







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  • Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
    – DonAntonio
    32 mins ago







  • 1




    Related : math.stackexchange.com/questions/833962/…
    – Arnaud D.
    19 mins ago










  • See also math.stackexchange.com/questions/1558734/…
    – Arnaud D.
    13 mins ago














up vote
4
down vote

favorite
2












I was wondering how many definitions of exponential functions can we think of. The basic ones could be:



$$e^x:=sum_k=0^inftyfracx^kk!$$
also
$$e^x:=lim_ntoinftybigg(1+fracxnbigg)^n$$
or this one:
Define $e^x:mathbbRrightarrowmathbbR\$ as unique function satisfying:
beginalign
e^xgeq x+1\
forall x,yinmathbbR:e^x+y=e^xe^y
endalign
Can anyone come up with something unusual? (Possibly with some explanation or references).






real-analysis definition big-list







share|cite|improve this question























  • Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
    – DonAntonio
    32 mins ago







  • 1




    Related : math.stackexchange.com/questions/833962/…
    – Arnaud D.
    19 mins ago










  • See also math.stackexchange.com/questions/1558734/…
    – Arnaud D.
    13 mins ago












up vote
4
down vote

favorite
2









up vote
4
down vote

favorite
2






2





I was wondering how many definitions of exponential functions can we think of. The basic ones could be:



$$e^x:=sum_k=0^inftyfracx^kk!$$
also
$$e^x:=lim_ntoinftybigg(1+fracxnbigg)^n$$
or this one:
Define $e^x:mathbbRrightarrowmathbbR\$ as unique function satisfying:
beginalign
e^xgeq x+1\
forall x,yinmathbbR:e^x+y=e^xe^y
endalign
Can anyone come up with something unusual? (Possibly with some explanation or references).






real-analysis definition big-list







share|cite|improve this question















I was wondering how many definitions of exponential functions can we think of. The basic ones could be:



$$e^x:=sum_k=0^inftyfracx^kk!$$
also
$$e^x:=lim_ntoinftybigg(1+fracxnbigg)^n$$
or this one:
Define $e^x:mathbbRrightarrowmathbbR\$ as unique function satisfying:
beginalign
e^xgeq x+1\
forall x,yinmathbbR:e^x+y=e^xe^y
endalign
Can anyone come up with something unusual? (Possibly with some explanation or references).






real-analysis definition big-list




real-analysis definition big-list






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 28 mins ago

























asked 33 mins ago









Michal Dvořák

764113




764113











  • Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
    – DonAntonio
    32 mins ago







  • 1




    Related : math.stackexchange.com/questions/833962/…
    – Arnaud D.
    19 mins ago










  • See also math.stackexchange.com/questions/1558734/…
    – Arnaud D.
    13 mins ago
















  • Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
    – DonAntonio
    32 mins ago







  • 1




    Related : math.stackexchange.com/questions/833962/…
    – Arnaud D.
    19 mins ago










  • See also math.stackexchange.com/questions/1558734/…
    – Arnaud D.
    13 mins ago















Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
– DonAntonio
32 mins ago





Your second definition is incorrect: the second $;1;$ must be $;x;$ , and the last one just doesn't make sense as it doesn't define anything at all...
– DonAntonio
32 mins ago





1




1




Related : math.stackexchange.com/questions/833962/…
– Arnaud D.
19 mins ago




Related : math.stackexchange.com/questions/833962/…
– Arnaud D.
19 mins ago












See also math.stackexchange.com/questions/1558734/…
– Arnaud D.
13 mins ago




See also math.stackexchange.com/questions/1558734/…
– Arnaud D.
13 mins ago










4 Answers
4






active

oldest

votes

















up vote
7
down vote













The exponential function is the unique solution of the initial value problem



$y'(x)=y(x) , quad y(0)=1$.






share|cite|improve this answer



























    up vote
    5
    down vote













    We can also define $e^x$ as



    • the inverse function of $ln x$ (defining $ln x$ independently)

    • the solution to $f'(x)=f(x)$ with $f(0)=1$





    share|cite|improve this answer



























      up vote
      0
      down vote













      The function which is invarient under differentiatin or integration and passing through $(0,1)$






      share|cite|improve this answer




















      • That's the same thing as Fred's answer (also mentioned by gimusi).
        – Arnaud D.
        21 mins ago

















      up vote
      0
      down vote













      Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.



      First define it for the natural numbers:



      Define $e^2 = e times e$, $e^3 = e times e times e $, etc.



      Now define it for other integers:



      $e^0 = 1$, $e^-n = frac1e^n$, etc.



      Now for other rational numbers (getting a bit harder):



      $e^fracpq = sqrt[q]e^p$



      Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.



      This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.






      share|cite|improve this answer






















      • Can you elaborate your answer with something concrete?
        – Michal Dvořák
        25 mins ago










      • I expanded my answer.
        – badjohn
        14 mins ago










      Your Answer





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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      7
      down vote













      The exponential function is the unique solution of the initial value problem



      $y'(x)=y(x) , quad y(0)=1$.






      share|cite|improve this answer
























        up vote
        7
        down vote













        The exponential function is the unique solution of the initial value problem



        $y'(x)=y(x) , quad y(0)=1$.






        share|cite|improve this answer






















          up vote
          7
          down vote










          up vote
          7
          down vote









          The exponential function is the unique solution of the initial value problem



          $y'(x)=y(x) , quad y(0)=1$.






          share|cite|improve this answer












          The exponential function is the unique solution of the initial value problem



          $y'(x)=y(x) , quad y(0)=1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 29 mins ago









          Fred

          40.8k1540




          40.8k1540




















              up vote
              5
              down vote













              We can also define $e^x$ as



              • the inverse function of $ln x$ (defining $ln x$ independently)

              • the solution to $f'(x)=f(x)$ with $f(0)=1$





              share|cite|improve this answer
























                up vote
                5
                down vote













                We can also define $e^x$ as



                • the inverse function of $ln x$ (defining $ln x$ independently)

                • the solution to $f'(x)=f(x)$ with $f(0)=1$





                share|cite|improve this answer






















                  up vote
                  5
                  down vote










                  up vote
                  5
                  down vote









                  We can also define $e^x$ as



                  • the inverse function of $ln x$ (defining $ln x$ independently)

                  • the solution to $f'(x)=f(x)$ with $f(0)=1$





                  share|cite|improve this answer












                  We can also define $e^x$ as



                  • the inverse function of $ln x$ (defining $ln x$ independently)

                  • the solution to $f'(x)=f(x)$ with $f(0)=1$






                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 29 mins ago









                  gimusi

                  82.3k74090




                  82.3k74090




















                      up vote
                      0
                      down vote













                      The function which is invarient under differentiatin or integration and passing through $(0,1)$






                      share|cite|improve this answer




















                      • That's the same thing as Fred's answer (also mentioned by gimusi).
                        – Arnaud D.
                        21 mins ago














                      up vote
                      0
                      down vote













                      The function which is invarient under differentiatin or integration and passing through $(0,1)$






                      share|cite|improve this answer




















                      • That's the same thing as Fred's answer (also mentioned by gimusi).
                        – Arnaud D.
                        21 mins ago












                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      The function which is invarient under differentiatin or integration and passing through $(0,1)$






                      share|cite|improve this answer












                      The function which is invarient under differentiatin or integration and passing through $(0,1)$







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered 22 mins ago









                      Mohammad Riazi-Kermani

                      38.2k41957




                      38.2k41957











                      • That's the same thing as Fred's answer (also mentioned by gimusi).
                        – Arnaud D.
                        21 mins ago
















                      • That's the same thing as Fred's answer (also mentioned by gimusi).
                        – Arnaud D.
                        21 mins ago















                      That's the same thing as Fred's answer (also mentioned by gimusi).
                      – Arnaud D.
                      21 mins ago




                      That's the same thing as Fred's answer (also mentioned by gimusi).
                      – Arnaud D.
                      21 mins ago










                      up vote
                      0
                      down vote













                      Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.



                      First define it for the natural numbers:



                      Define $e^2 = e times e$, $e^3 = e times e times e $, etc.



                      Now define it for other integers:



                      $e^0 = 1$, $e^-n = frac1e^n$, etc.



                      Now for other rational numbers (getting a bit harder):



                      $e^fracpq = sqrt[q]e^p$



                      Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.



                      This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.






                      share|cite|improve this answer






















                      • Can you elaborate your answer with something concrete?
                        – Michal Dvořák
                        25 mins ago










                      • I expanded my answer.
                        – badjohn
                        14 mins ago














                      up vote
                      0
                      down vote













                      Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.



                      First define it for the natural numbers:



                      Define $e^2 = e times e$, $e^3 = e times e times e $, etc.



                      Now define it for other integers:



                      $e^0 = 1$, $e^-n = frac1e^n$, etc.



                      Now for other rational numbers (getting a bit harder):



                      $e^fracpq = sqrt[q]e^p$



                      Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.



                      This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.






                      share|cite|improve this answer






















                      • Can you elaborate your answer with something concrete?
                        – Michal Dvořák
                        25 mins ago










                      • I expanded my answer.
                        – badjohn
                        14 mins ago












                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.



                      First define it for the natural numbers:



                      Define $e^2 = e times e$, $e^3 = e times e times e $, etc.



                      Now define it for other integers:



                      $e^0 = 1$, $e^-n = frac1e^n$, etc.



                      Now for other rational numbers (getting a bit harder):



                      $e^fracpq = sqrt[q]e^p$



                      Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.



                      This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.






                      share|cite|improve this answer














                      Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.



                      First define it for the natural numbers:



                      Define $e^2 = e times e$, $e^3 = e times e times e $, etc.



                      Now define it for other integers:



                      $e^0 = 1$, $e^-n = frac1e^n$, etc.



                      Now for other rational numbers (getting a bit harder):



                      $e^fracpq = sqrt[q]e^p$



                      Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.



                      This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 14 mins ago

























                      answered 26 mins ago









                      badjohn

                      3,6291618




                      3,6291618











                      • Can you elaborate your answer with something concrete?
                        – Michal Dvořák
                        25 mins ago










                      • I expanded my answer.
                        – badjohn
                        14 mins ago
















                      • Can you elaborate your answer with something concrete?
                        – Michal Dvořák
                        25 mins ago










                      • I expanded my answer.
                        – badjohn
                        14 mins ago















                      Can you elaborate your answer with something concrete?
                      – Michal Dvořák
                      25 mins ago




                      Can you elaborate your answer with something concrete?
                      – Michal Dvořák
                      25 mins ago












                      I expanded my answer.
                      – badjohn
                      14 mins ago




                      I expanded my answer.
                      – badjohn
                      14 mins ago

















                       

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