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I^i = e^(-pie/2) in physics?

  1. Apr 11, 2006 #1
    Is this identity where i=srt(-1) used at all in physics or has a physical interpretation? It certinaly is very unintuitive for me even as a mathematical concept although I see how it is derived.
  2. jcsd
  3. Apr 11, 2006 #2
    the identity [tex] i^{i}=e^{-\pi/2} [/tex] is mathematically valid so it should also be valid in physics although i can,t remember any particular example...
  4. Apr 11, 2006 #3


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    That exponetial crops up all over the place but the abstract nature of i to the power i prohibits anyone from using it instead of the exponential. I've never seen a derivation that included such a substitution before though.
  5. Apr 11, 2006 #4


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    Must...Resist..."Pie" cheap shot.
  6. Apr 11, 2006 #5


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    All the symbols (e,i,pi,-,2,/) in the equation are defined in mathematical terms. They are used by physicists, but they have no physical meaning.
  7. Apr 12, 2006 #6


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    By computing various contour integrals in the complex plane and invoking the residue theorem, I'm sure that at one point or the other, some physicist has implicitly used that result when switching from one path of integration to another.
  8. Apr 12, 2006 #7
    Intuition can be both amazinglu helpful and deadly dangourous in Mathematics, Axiom of Choice being the prime example. ANyway, equations of pure mathematics have no direct "interpretation" in physics. It's only when you decide to define the various involved quanitites to have physical meanings do the equation itself gain a physical interpretation.

    Whether it's used in physics is a valid question however, I don't know the answer.

  9. Apr 12, 2006 #8


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    don't think i would agree with that. perhaps there are no actual physical quantities that are imaginary (so complex math is a useful abstraction in describing some physical phenomena and might not be directly meaningful physically), but "2" and "pi" and operations like "-" or "+" definitely have physical meaning.
  10. Apr 12, 2006 #9
    You have to have to allow for the fact that the phase of a complex number plus any integer multiple of 2pi is the same complex number. So

    [tex]i^i=e^{-(\tfrac{\pi}{2}+2n\pi )}[/tex]

    And no I've never seen it used.
  11. Apr 12, 2006 #10
    Yes, this is very much used in physics. this all is very important in electrical science, when u go for the working of circuits containing capacitors, inductors and resistors. And this complex variable has a complete complex mathematics with it much like our real mathematics or u can say the one u would have done till date. In later stages or at graduate level u will know about a different branch of mathematics containing everything like ordinary maths but has complex variable attached to it.
  12. Apr 12, 2006 #11


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    Physicists do use complex functions of a real variable often (in wave mechanics or electrical circuits), but functions from the complex plane to the complex plane I haven't encountered yet. The independent variables are usually time or space variables.
  13. Apr 13, 2006 #12
    I agree with this. I'd even go further to say that "2", "pi", "-", "+" were first invented or thought up in the real world, in a physically intuitive context when these things were needed to do something.

    Only later were they defined to be mathematical objects. Then even later did the theoretical physicists start to borrow these mathematical entities and attach formal physical interpretations to them. Although the ordering of the latter two may change when a Witten or a Feynman comes along but generally it is this way.
    Last edited: Apr 13, 2006
  14. Apr 13, 2006 #13
    I was looking for a more specific physical interpretation for this identity but looks like it is not directly used or defined like the Born Interpretation but is used in a physical derivation or calculation.

    This raises another question which is, when doing a calculation in physics and going from one equation to another and finally to an answer which is correct. It would mean that the mathematical equations at the start of the problem had a physical interpretation and the equations at the end of the calculation also has a physical interpretaion. This should imply each step in your calculation should be able to have a physical interpretation attached as well (i.e. it should be possible to find one no matter how unnatural it is to do so). If it was not the case than when does the physical interpretation dissapear and reappear in your calculation?
    Therefore, if this identity is used somewhere in the mathematics in a physical problem, this identity should have attached a physical meaning. ALthough in most cases this meaning is forced and unnatural.

    But my claim could be wrong in the first place. Maybe it is the case that by followinw the rules of defined mathematics, hence by following tautological steps, the physical meaning is never lost in the equations, no matter which step you are at.

    But no matter which way you go, it would suggest that if this identity is used somewhere in physics than it has a physical meaning however vague it may be.
    Last edited: Apr 13, 2006
  15. Apr 14, 2006 #14


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    even though the fundamental identity:

    [tex] e^{i \theta} = \cos(\theta) + i \sin(\theta) [/tex]

    and variants is used extensively in physics and engineering (and can be given some physical meaning even if there are really no complex physical quantities out there) but

    [tex] i^i = e^{i \log(i)} = e^{i (i \pi/2)} = e^{-\pi/2} [/tex]

    is used nowhere in physics lit that i have seen. but what do i know?
  16. Apr 14, 2006 #15
    It is not necessary for all intermediate steps to have direct representations in physical reality. For example, all quantum physicists use the wave function in their calculations, but not all acknowledge it to be a "real" entity. It's treated as a calculational device by them.
  17. Apr 14, 2006 #16
    No physical meaning - imaginary or fundermental

    They are our best guess, so perhpas we should start to learn to accept them for what they are and move forward based on the information and knowledge we have already gained.

    We live and think in a 4 dimensional universe and cannot understand or explain what it would be like to see from 5 Dimensions only in abstract mathematics and physics, which only a handful of people understand.

    The world becomes a much simplier place if you try to look from above in 5 dimensions, downwards.

    Why not explain it and describe it to everyone in simple plain english or diagrams and ask for any new suggestions or alternative ideas.

    Who said "imagination is more important than knowledge", you might be suprized with the results, some might even be useful


    Terry Giblin
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