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Math Help - Complex Analysis - Mapping of a straight line and Principal value

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    Complex Analysis - Mapping of a straight line and Principal value



    I've done part a) I just simply cubed z and expanded, then seperated real and imaginary. I assume this is all that's needed?

    For part b) I have no idea how to do it.
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    Quote Originally Posted by fourierT View Post


    I've done part a) I just simply cubed z and expanded, then seperated real and imaginary. I assume this is all that's needed?

    For part b) I have no idea how to do it.
    I doubt they are talking about the Cauchy Principal value so
    what I'm going to assume they mean in (b) is simply the value of \small 2^{i+1} \text { for } (0 \leq \theta <2\pi)

    \small 2^{i+1}=e^{(i+1)ln(2)}

    \text{You can probably come up with } z=r e^{i\theta}\text{ for that.}
    Last edited by romsek; December 11th 2013 at 06:01 PM.
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    \small 2^{i+1}=e^{(i+1)ln(2)}

    \text{You can probably come up with } z=r e^{i\theta}\text{ for that.}

    Is it cos(ln2) + (i+1)sin(ln2)

    ?
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    Quote Originally Posted by fourierT View Post
    \small 2^{i+1}=e^{(i+1)ln(2)}

    \text{You can probably come up with } z=r e^{i\theta}\text{ for that.}

    Is it cos(ln2) + (i+1)sin(ln2)

    ?
    no...

    e^{(i+1)ln(2)} = e^{ln(2) + ln(2) i} = 2 * e^{ln(2) i} = 2(cos(ln(2))+i sin(ln(2)))

    you could have started with

    2^{i+1}=2*2^{i}=2e^{ln(2)i}

    I guess that's a bit quicker
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    Quote Originally Posted by romsek View Post
    no...

    e^{(i+1)ln(2)} = e^{ln(2) + ln(2) i} = 2 * e^{ln(2) i} = 2(cos(ln(2))+i sin(ln(2)))

    you could have started with

    2^{i+1}=2*2^{i}=2e^{ln(2)i}

    I guess that's a bit quicker
    How can we evaluate that in algebraic form..wouldn't it just be numbers.
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    Quote Originally Posted by fourierT View Post
    How can we evaluate that in algebraic form..wouldn't it just be numbers.
    you see any variables in 2^{i+1} ?
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    Sure, i is algebraic, but is that all it wants, the real part and the imaginary part?
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    Re: Complex Analysis - Mapping of a straight line and Principal value

    the algebraic form of a complex number is z = x + i y

    that's all it is.
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