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  1. #1
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    exponential

    hi can someone please explain to me how u evaluate  e^{2 \pi i} And  e^{i \pi} .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
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  2. #2
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    Quote Originally Posted by Dgphru View Post
    hi can someone please explain to me how u evaluate  e^{2 \pi i} And  e^{i \pi} .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
    Let \varphi be a real number, then e^{i\cdot\varphi} is a point on the unit-circle (radius 1, center at 0) in the complex plane, where \varphi is the angle in radians corresponding to that point as seen from the origin 0, and relative to the positive direction of the real axis.
    Because of this, the real part of that complex number e^{i\cdot\varphi} is \cos(\varphi), and its imaginary part is \sin(\varphi).
    Overall, you get the relationship e^{i\cdot\varphi}=\cos(\varphi)+i\cdot\sin(\varphi  ).
    (See also: Euler's formula - Wikipedia, the free encyclopedia).

    What you get, therefore, is in the case of e^{2 \pi i}=\cos(2\pi)+i\cdot\sin(2\pi)=1+i\cdot 0=1
    and in the case of e^{i\cdot \pi}=\cos(\pi)+i\cdot\sin(\pi)=-1+i\cdot 0=-1.
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  3. #3
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    Quote Originally Posted by Dgphru View Post
    hi can someone please explain to me how u evaluate  e^{2 \pi i} And  e^{i \pi} .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
    Euler's formula states that:
    e^(ix) = cos x + i sin x
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  4. #4
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    Quote Originally Posted by Dgphru View Post
    hi can someone please explain to me how u evaluate  e^{2 \pi i} And  e^{i \pi} .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
    I can't not answer this one

    e^{ix} = \cos(x) + i\sin(x)

    Therefore e^{i\pi} = \cos \pi + i \sin \pi = -1

    By the same logic e^{2\pi i} = \cos (2\pi) + i \sin (2\pi) = 1

    For any integer k: e^{k\pi i} = \cos (k\pi) = \pm 1 depending on if k is odd or even
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