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Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]

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  • Shlomi Fish
    ... Actually, IINM, e^(i*t)=cos(t)+i*sin(t) is more of a definition then a fact. I believe Euler noticed that the sinousidal functions behaved very much like
    Message 1 of 9 , Feb 7 12:12 AM
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      >Here's some very interesting reading on mathematics, discussing the
      >question: Why should simple mathematical techniques and constants so
      >perfectly describe natural phenomena? I've often wondered why e and pi
      >turn up in descriptions of all sorts of phenomena that, on their face,
      >have nothing to do with logarithms or geometry. One of my favorite
      >identities, with almost the force of a koan, is e ** (-i * pi) = -1.
      >It's just not reasonable that e, complex numbers, and circles should be
      >so intimately related to one another.

      Actually, IINM, e^(i*t)=cos(t)+i*sin(t) is more of a definition then a
      fact. I believe Euler noticed that the sinousidal functions behaved very
      much like exponents so he decided to extend the definition of the function
      e^x that way. I don't think there's really a proof that it is indeed so.

      Indeed, one can show that all the usual exponentation operations behave
      the same for it too, but it's possible that there are other possible ways
      to extend it. Or maybe not, considering the fact that the function e^z is
      a function from the complex plane to the complex plane.

      For more information, Technion students may wish to take the course
      Complex Functions, which I believe has some more serious equivalents from
      the Math faculty point-of-view. Then again, they may not, because although
      interesting, it's not a breath-taking course.

      Regards,

      Shlomi Fish


      ----------------------------------------------------------------------
      Shlomi Fish shlomif@...
      Home Page: http://t2.technion.ac.il/~shlomif/
      Home E-mail: shlomif@...

      The prefix "God Said" has the extraordinary logical property of
      converting any statement that follows it into a true one.
    • Nadav Har'El
      On Wed, Feb 07, 2001, Shlomi Fish wrote about Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics] : First time I quote the egroups ad in my
      Message 2 of 9 , Feb 7 2:04 AM
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        On Wed, Feb 07, 2001, Shlomi Fish wrote about "Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]":

        First time I quote the egroups ad in my reply!

        So now egroups is yahoo-groups? Ooof, now I need to change my mail filtering
        scripts in about a dozen places ;)

        --
        Nadav Har'El | Wednesday, Feb 7 2001, 14 Shevat 5761
        nyh@... |-----------------------------------------
        Phone: +972-53-245868, ICQ 13349191 |I want to be a human being, not a human
        http://nadav.harel.org.il |doing -- Scatman John
      • Nadav Har'El
        ... Not exactly :) It is not some sort of arbitrary definition... If you write down the taylor series for e^x, and fix that for a definition, then when you
        Message 3 of 9 , Feb 7 2:20 AM
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          On Wed, Feb 07, 2001, Shlomi Fish wrote about "Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]":
          > Actually, IINM, e^(i*t)=cos(t)+i*sin(t) is more of a definition then a
          > fact.
          > I believe Euler noticed that the sinousidal functions behaved very
          > much like exponents so he decided to extend the definition of the function
          > e^x that way. I don't think there's really a proof that it is indeed so.

          Not exactly :) It is not some sort of arbitrary definition... If you write
          down the taylor series for e^x, and fix that for a definition, then when you
          substitute complex x's, you'll automatically get the cos(t)+isin(t)
          definition for complex arguments.

          I'm sure there are other arguments why e(z), or more generally, "chezkot",
          "must" be defined this way, I just don't remember...

          > Indeed, one can show that all the usual exponentation operations behave
          > the same for it too, but it's possible that there are other possible ways
          > to extend it. Or maybe not, considering the fact that the function e^z is
          > a function from the complex plane to the complex plane.

          If I remember correctly, e^z (as "defined" above) is the only complex
          extention of e^x. But it has been many years since I studied complex functions
          (called "function theory" in the Math department at the Technion).

          > For more information, Technion students may wish to take the course
          > Complex Functions, which I believe has some more serious equivalents from
          > the Math faculty point-of-view. Then again, they may not, because although
          > interesting, it's not a breath-taking course.

          :)

          --
          Nadav Har'El | Wednesday, Feb 7 2001, 14 Shevat 5761
          nyh@... |-----------------------------------------
          Phone: +972-53-245868, ICQ 13349191 |I want to live forever or die in the
          http://nadav.harel.org.il |attempt.
        • Nadav Har'El
          ... And now I m replying to my own message :) I just wanted to point out that egroups (aka yahoo-groups) removed the ad I quoted in my reply, so my reply ended
          Message 4 of 9 , Feb 7 2:23 AM
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            On Wed, Feb 07, 2001, To hackers-il@yahoogroups.com wrote about "Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]":
            > On Wed, Feb 07, 2001, Shlomi Fish wrote about "Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]":
            >
            > First time I quote the egroups ad in my reply!

            And now I'm replying to my own message :) I just wanted to point out that
            egroups (aka yahoo-groups) removed the ad I quoted in my reply, so my reply
            ended up quoting nothing. How quaint...


            --
            Nadav Har'El | Wednesday, Feb 7 2001, 14 Shevat 5761
            nyh@... |-----------------------------------------
            Phone: +972-53-245868, ICQ 13349191 |For people who like peace and quiet - a
            http://nadav.harel.org.il |phoneless cord.
          • Moshe Zadka
            ... Depends a lot of what the definition of e^x is. It s no good to say the usual power definition , because usually, a^b is defined as e^(ln(a)*b) . The
            Message 5 of 9 , Feb 8 8:49 AM
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              On Wed, 7 Feb 2001, Shlomi Fish <shlomif@...> wrote:

              > Actually, IINM, e^(i*t)=cos(t)+i*sin(t) is more of a definition then a
              > fact. I believe Euler noticed that the sinousidal functions behaved very
              > much like exponents so he decided to extend the definition of the function
              > e^x that way. I don't think there's really a proof that it is indeed so.

              Depends a lot of what the definition of e^x is. It's no good to say
              "the usual power definition", because usually, "a^b" is defined as
              "e^(ln(a)*b)".

              The usual definition is
              oo
              e^x = sigma x^n*(1/n!)
              n=0

              In that case, it is a theorem that e^x=cos(x)+i*sin(x)
              --
              For public key: finger moshez@... | gpg --import
              <doogie> Debian - All the power, without the silly hat.
            • Oleg Goldshmidt
              ... Actually, the usual definition is lim (1+1/n)^(x*n) as n- oo ;-). Of course, you are right that exp can be defined through the series, but a lot of
              Message 6 of 9 , Feb 8 9:11 AM
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                Moshe Zadka <moshez@...> writes:

                > The usual definition is
                > oo
                > e^x = sigma x^n*(1/n!)
                > n=0

                Actually, the "usual" definition is lim (1+1/n)^(x*n) as n->oo ;-).

                Of course, you are right that exp can be defined through the
                series, but a lot of logical steps become hidden, such as analyticity
                of the function, which guarantees that the Taylor expansion is unique,
                and thus suitable for defining exp.

                > In that case, it is a theorem that e^x=cos(x)+i*sin(x)

                I think the proof (in a few words) was posted by someone here
                (Nadav?): the MacLauren series (being identical to the functions, as
                they are analytical) for exp, sin, and cos lead to the Euler
                identity.

                Dash! My complex analysis is rusty - back to books? ;-)

                --
                Oleg Goldshmidt | ogoldshmidt@...
                "... We work by wit, and not by witchcraft;
                And wit depends on dilatory time." [Shakespeare]
              • Nadav Har'El
                ... Don t laugh, but many times teachers of Infi 1 indeed define e in such a way. Why? Think about it: we want to define e as the base such that f(x)=e^x
                Message 7 of 9 , Feb 8 2:05 PM
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                  On Thu, Feb 08, 2001, Oleg Goldshmidt wrote about "Re: [hackers-il] [Fwd: The Unreasonable Effectiveness of Mathematics]":
                  > Moshe Zadka <moshez@...> writes:
                  >
                  > > The usual definition is
                  > > oo
                  > > e^x = sigma x^n*(1/n!)
                  > > n=0
                  >
                  > Actually, the "usual" definition is lim (1+1/n)^(x*n) as n->oo ;-).

                  Don't laugh, but many times teachers of Infi 1 indeed define "e" in such
                  a way. Why? Think about it: we want to define e as the base such that
                  f(x)=e^x gives f'=f (or equivalently, such that log_e'(x)=1/x).
                  It's easy to define such an f with a taylor series (see more on that below),
                  but you learn about e^x before you learn Taylor series! Another way to
                  define e^x, by defining log_e as integral of 1/x, is also impossible before
                  you learn what an integral is (and in the Technion math department, you don't
                  learn integration until the second semester). So, you make definitions like
                  the one you showed above, and go ahead proving, using the basic definitions
                  and epsilons and deltas, that (e^x)'=(e^x) or (log_e)=1/x. I don't remember
                  all the details, but you may find them in some Infi books (though not all
                  of them go this way). I think (but this was 10 years ago, so I might be
                  wrong) that this is they way we were taught e^x.

                  But such a defintion cannot be extended to define e^z where z is complex,
                  because you'd still need to define how to raise a number to a complex power,
                  so it's a chicken-and-egg problem. Once you define a Taylor series, and
                  show e^x's Taylor series, you notice that it's very easy to extend the
                  taylor series to a wider selection of x's: you can substitute complex z's
                  instead of the x's, and can even substitute other crazy things, such as
                  whole matrices, and obtain a defintion for e^A, where A is a matrix.
                  Anybody who learned ODE (ordinary differential equations) will probably
                  know this fact.

                  By the way, if you think that defining the complex e^z by its Taylor series
                  is strange, awkward, or ad-hoc, then try thinking for a moment why e^x is
                  used at all, and not, say, 10^x or 2^x? The answer is that e^x is the only
                  base that results in a function f=f', and that (together with e^0=1)
                  immediately results in e^x's familiar Taylor series. So, if you want a
                  function f(z) whose f'(z)=f(z) even for complex numbers, then you'll need
                  to define it using that same taylor series.

                  The e^A for matrices I mentioned above is useful for the same reason:
                  (e^A)'=e^A (where derivative is defined as expected), and that is true
                  exactly because of the taylor series used to define it.

                  > Of course, you are right that exp can be defined through the
                  > series, but a lot of logical steps become hidden, such as analyticity
                  > of the function, which guarantees that the Taylor expansion is unique,
                  > and thus suitable for defining exp.

                  I don't remember the details (which is, frankly, quite shameful...), this
                  logic seems a bit circular. You can't say e^z is "inherently analytic"
                  before you actually define it. What you can do is say that e^x must have
                  a unique analytic extension, and that the taylor series is indeed one of
                  them (1. analytic and 2. an extension) and thus the only possible one.
                  But I don't remmeber the details, or the order in which teachers usually
                  build the definition of e^z. Correct me if I'm wrong - it has been many
                  years since I last touched this subject :(

                  --
                  Nadav Har'El | Thursday, Feb 8 2001, 16 Shevat 5761
                  nyh@... |-----------------------------------------
                  Phone: +972-53-245868, ICQ 13349191 |If I were two-faced, would I be wearing
                  http://nadav.harel.org.il |this one?.... Abraham Lincoln
                • Shlomi Fish
                  ... It reminds of a joke I saw on a blacboard in the Technion: Technion = Integral from tau=-infinity to t of Hell Hell = Integral from tau=-infinity to t of
                  Message 8 of 9 , Feb 11 8:57 AM
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                    On Fri, 9 Feb 2001, Nadav Har'El wrote:

                    > Snipped...
                    >
                    >
                    >

                    It reminds of a joke I saw on a blacboard in the Technion:

                    Technion = Integral from tau=-infinity to t of Hell
                    Hell = Integral from tau=-infinity to t of Technion

                    Conclusion: Technion = Hell = e^t

                    And here's another one I saw written on a desk on Fishbach. It only works
                    in Hebrew so here goes:

                    Haderekh Aruka Umefutelth.
                    Ani Oseh Matam,
                    Oseh Matam,
                    Oseh Matam

                    Gam im hahavah bo'ereth
                    Ani Oseh Matam Berosh Muram...

                    Regards,

                    Shlomi Fish


                    ----------------------------------------------------------------------
                    Shlomi Fish shlomif@...
                    Home Page: http://t2.technion.ac.il/~shlomif/
                    Home E-mail: shlomif@...

                    The prefix "God Said" has the extraordinary logical property of
                    converting any statement that follows it into a true one.
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