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Solve equation with unknown x

  1. Nov 2, 2015 #1
    1. The problem statement, all variables and given/known data
    I have to solve the following equation:
    Screenshot_1.jpg

    2. Relevant equations


    3. The attempt at a solution
    I know that since the right side is 1 and on the left side i have i (imaginary number) it means that i could rewrite right side as cos0 + isin0 since it's the same, but what can i do with left side, it obviously can't stay like this because of the product, i have to get rid of it, i just don't have an idea.
     
  2. jcsd
  3. Nov 2, 2015 #2

    Samy_A

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    What do you get when you replace the factors in the product using Euler's formula?
     
  4. Nov 2, 2015 #3
    As far as i know, we have Пekx (П is from 1 to n) which is equal to:

    ex*e2x*...*enx or

    e(1+2+...+n)x. Now, i am not 100% sure, but i believe is should represent 1+2+...+n as n(n+1)/2 and go back to the trigonometric form of number, and then simply find x since i know exact value of angle for the right side. Am i correct?
     
  5. Nov 2, 2015 #4

    Samy_A

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    You forgot the ##i## in your exponent, as Euler's formula is ##cos(y)+isin(y)=e^{iy}##.
    You are correct about ##\sum_{k=1}^n k =n(n+1)/2##.
    As the product is equal to ##1##, you can now indeed solve for x by applying Euler's formula again: the cosine of your exponent (not including the ##i##) must be ##1##.
     
    Last edited: Nov 2, 2015
  6. Nov 2, 2015 #5

    RUber

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    How many solutions do you need? Just one, all of them for a fixed n, or all of them for any n?
    ##e^{iy} = 1 ## is true for an infinite number of periodic terms.
    For a large w, ##e^{iwy} ## will have a much shorter period. Your w might get quite large as n grows.
     
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