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Math Help - Understanding a formula

  1. #1
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    Understanding a formula

    i^-i= \sqrt{e^\pi}

    I read here, The number e, that this formula is not understood.

    "In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula above. In his lectures he would say to his students:-
    Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.:
    Is there a meaning to this equation?

    I tried to understand it with my calculator, but failed horribly.
    i^-i=  \sqrt{e^\pi} =about 4.787
    With my limited understanding of i, that makes absolutely sense to me whatsoever.

    I'm doing a research paper for undergraduate admissions into Cal Tech about 'e'.


    Thanks!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Truthbetold View Post
    i^-i= \sqrt{e^\pi}

    I read here, The number e, that this formula is not understood.

    "In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula above. In his lectures he would say to his students:-
    Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.:
    Is there a meaning to this equation?

    I tried to understand it with my calculator, but failed horribly.
    i^-i=  \sqrt{e^\pi} =about 4.787
    With my limited understanding of i, that makes absolutely sense to me whatsoever.

    I'm doing a research paper for undergraduate admissions into Cal Tech about 'e'.


    Thanks!
    are you familiar with Euler's formula? e^{i \theta} = \cos \theta + i \sin \theta.

    with that we can see that i = e^{i(\pi / 2 + 2 n \pi)} for any integer n. since all will give the same value, we can take one, e^{i \pi / 2} is the nicest.

    thus we have, i^{-i} = (e^{i \pi / 2})^{-i} = e^{\pi / 2} = (e^\pi)^{1/2} = \sqrt{e^\pi}
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