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Math Help - Find ln z?

  1. #1
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    Find ln z?

    ln z = ln (mod of z) + i Arg z +- (2n pi) i

    ln (e3i )
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    Re: Find ln z?

    Can't you immediately read off what \displaystyle \left| e^{3i} \right| and \displaystyle \arg{\left( e^{3i} \right)} are?
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    Re: Find ln z?

    I can get the mod but i dont how to find the Arg of e^3i, please teach me how to find the Arg of this kinda function
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    Re: Find ln z?

    Quote Originally Posted by szak1592 View Post
    I can get the mod but i dont how to find the Arg of e^3i, please teach me how to find the Arg of this kinda function
    e^z=e^{x+yi}=e^x[\cos(y)+i\sin(y)] so \text{arg}(e^z)=~?
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    Re: Find ln z?

    Quote Originally Posted by szak1592 View Post
    I can get the mod but i dont how to find the Arg of e^3i, please teach me how to find the Arg of this kinda function
    Surely you can see it's written as \displaystyle e^{\theta i}. What is \displaystyle \theta?
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    Re: Find ln z?

    okay it is 3, so cos 3 + i sin 3 so it means arg z = arc tan of sin3/cos3, rite???
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    Re: Find ln z?

    You are overcomplicating things. The argument is the angle that is made. You already read off that this angle is 3 (radians).

    Of course if you were going to do it your way, you'd note that sin(x)/cos(x) = tan(x) and so your arctangent cancels off anyway.
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    Re: Find ln z?

    ln (e3i )
    You should have noticed that the ln and e^ functions are inverses of each other, so in at least some branch, ln(e^(z)) = z. Then think about what the angle/magnitude of this would be, whether this result makes sense and if not, what you need to do to make it make sense.
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