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Math Help - possible values of imaginary numbers

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    possible values of imaginary numbers

    Two similar questions concerning imaginary numbers. What are all the possible values of [Math]\log(1+i)[/tex] and of i^{i}? Please help, due soon.
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    Quote Originally Posted by markholden View Post
    Two similar questions concerning imaginary numbers. What are all the possible values of [Math]\log(1+i)[/tex] and of i^{i}? Please help, due soon.
    i^i=e^{-\pi/2}
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  3. #3
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    Quote Originally Posted by markholden View Post
    Two similar questions concerning imaginary numbers. What are all the possible values of \log(1+i) and of i^{i}? Please help, due soon.
    Suppose y=\log(1+i), by this we mean that:

    e^y=1+i,

    Now suppose y=u+iv, then:

    e^u e^{iv}=1+i, and |e^u e^{iv}|=e^u=|1+i|=\sqrt{2},

    so u=\log(\sqrt{2}).

    Now:

    e^{iv}=\cos(v)+i \sin(v)=1/\sqrt{2}+i/\sqrt{2},

    so:

    <br />
v=\pi/4 + 2 \pi n,\ \ \ n=0,\ \pm 1,\ \pm 2,\ \ ...<br />

    Hence:

    <br />
y=\log (\sqrt{2})+\pi i (1/4 +2n),\ \ n=0,\ \pm 1,\ \pm 2,\ \ ...<br />

    RonL
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    Quote Originally Posted by ThePerfectHacker View Post
    i^i=e^{-\pi/2}
    <br />
i=e^{i \pi /2 + i 2 \pi n}, \ \ \ n=0, \pm 1,\ \pm 2, ..<br />

    so:

    <br />
i^i=e^{- \pi /2 - 2 \pi n}, \ \ \ n=0, \pm 1,\ \pm 2, ..<br />

    RonL
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