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Math Help - Complex number(logarithm)

  1. #1
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    Complex number(logarithm)

    Let z = 4e^i(pi/6)
    find iz and |e^iz|

    what is iz?
    is it imaginary part of z?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nameck View Post
    Let z = 4e^i(pi/6)
    find iz and |e^iz|

    what is iz?
    is it imaginary part of z?
    I would assume so. Can you do that part?

    For the second part note that

    \left|z\right|=\sqrt{\Re^2(z)+\Im^2(z)} and since \Re\left(e^{iz}\right)=\cos(z),\Im\left(e^{iz}\rig  ht)=\sin(z) it follows that \left|e^{iz}\right|=\sqrt{\sin^2(z)+\cos^2(z)}=1. Another way to realize this is that e^{iz} is some point \zeta on the unit circle. Thus \left|e^{iz}\right| is the length of the radius \overline{O\zeta}, but since this is the unit circle this is trivially one.
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  3. #3
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    got it for the second part.. what is the method to slove the first part?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nameck View Post
    got it for the second part.. what is the method to slove the first part?
    As I said \Im\left(e^{iz}\right)=\sin\left(z\right). What's z?
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  5. #5
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    Quote Originally Posted by nameck View Post
    Let z = 4e^i(pi/6)
    find iz and |e^iz|

    what is iz?
    is it imaginary part of z?
    That would not be my interpretation! iz is just what it looks like: i times z.

    You could either write z in "rectangular form", a+ bi, and then iz= -b+ ai, or write i in polar form, e^{i\pi/2} and multiply 4e^{i\pi/6}e^{i\pi/2}.
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