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Disprooving Euler

  1. Jul 3, 2012 #1
    I was messing around with Euler’s Identity and I think I accidently disproved it. I would like someone to check my math to make sure I didn’t make any rookie mistakes.

    e^{\pi i} + 1 = 0 \\
    e^{\pi i} = - 1 \\
    \left( {e^{\pi i} } \right)^2 = \left( { - 1} \right)^2 \\
    e^{2\pi i} = 1 \\
    \ln \left( {e^{2\pi i} } \right) = \ln \left( 1 \right) \\
    2\pi i = 0 \\
    \frac{{2\pi i}}{{\pi i}} = \frac{0}{{\pi i}} \\
    2 = 0 \\
  2. jcsd
  3. Jul 3, 2012 #2


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    So, how did you define the logarithm on complex numbers??
  4. Jul 3, 2012 #3


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    ln(1) = 2kπi, where k is any integer. ln is a multivalued function.
  5. Jul 3, 2012 #4

    What gives you away as a rookie is the title of your post, not your mathematics...which are also wrong.

    Perhaps you'll be interested in reading about the complex logarithmic function's definition...

  6. Jul 3, 2012 #5


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    edit: student posted under my account
  7. Jul 3, 2012 #6
    Thank you, DonAntonio, mathman and micromass, I thought that was what my error was, but I wasn't sure. You see, I haven't yet taken a course in which I learn even the basics of complex numbers, so my knowledge in that area is rather lacking.
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