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Math Help - The Complex Exponential And Regions in the Complex Plane

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    The Complex Exponential And Regions in the Complex Plane

    Hello all, I have done a few questions that I need corrected.

    1.) Sketch the region of the complex plane defined by Im (z^2) = 2 for pi/8 < arg (z) < 3pi/8

    my solution: I let z = x + iy, then found z^2 and the imaginary parts of z^2 which gave me a graph shape of 1/x. the region defined by the angle is in the first quadrant and only a small bit of that curve is included.

    __________________________________________________ ______________________

    2.) z = 1/4 - [root(3) / 4]i

    a) Write z in exponential polar form for angles between -pi and pi

    my solution: 1/2 * e ^ i (pi/3)

    b) Calculate (1/z) ^ 9

    my solution: -2 * 2^8 + 0i

    __________________________________________________ ______________________

    3.) Use the complex exponential to express sin^3(theta) as a sum of sines or cosines of multiples of theta.

    My solution: -1/i sin(3 theta) + 3/i cos (theta)
    __________________________________________________ ______________________

    Thank you in advance.
    Last edited by andrew2322; September 10th 2011 at 02:48 AM. Reason: Im(z^2) = 2
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    Re: The Complex Exponential And Regions in the Complex Plane

    Quote Originally Posted by andrew2322 View Post
    Hello all, I have done a few questions that I need corrected.

    1.) Sketch the region of the complex plane defined by Im (z^2) for pi/8 < arg (z) < 3pi/8

    my solution: I let z = x + iy, then found z^2 and the imaginary parts of z^2 which gave me a graph shape of 1/x. the region defined by the angle is in the first quadrant and only a small bit of that curve is included.

    __________________________________________________ ______________________

    2.) z = 1/4 - [root(3) / 4]i

    a) Write z in exponential polar form for angles between -pi and pi

    my solution: 1/2 * e ^ i (pi/3)

    b) Calculate (1/z) ^ 9

    my solution: -2 * 2^8 + 0i

    __________________________________________________ ______________________

    3.) Use the complex exponential to express sin^3(theta) as a sum of sines or cosines of multiples of theta.

    My solution: -1/i sin(3 theta) + 3/i cos (theta)
    __________________________________________________ ______________________

    Thank you in advance.
    1. Are you sure it didn't say \displaystyle \frac{\pi}{8} < \arg{\left(z^2\right)} < \frac{3\pi}{8}

    2. a) It's clear that \displaystyle z = \frac{1}{4} - \frac{\sqrt{3}}{4}i is in the fourth quadrant, so \displaystyle \arg{(z)} can not possibly be \displaystyle \frac{\pi}{3}.

    b) This is easiest to evaluate if you have evaluated part a) correctly...
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    Re: The Complex Exponential And Regions in the Complex Plane

    Hello Prove it. I checked the question again and it says arg (z) not arg (z^2)

    also, for part 2a) I realise my careless mistake, I was meant to type - pi/3.

    due apologies.
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    Re: The Complex Exponential And Regions in the Complex Plane

    Sorry Prove It and others. The Im (z^2) = 2, i forgot to include the = 2 part. very sorry.

    I have edited the original post to include this correction.
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    Re: The Complex Exponential And Regions in the Complex Plane

    Quote Originally Posted by andrew2322 View Post
    Hello Prove it. I checked the question again and it says arg (z) not arg (z^2)

    also, for part 2a) I realise my careless mistake, I was meant to type - pi/3.

    due apologies.
    Then \displaystyle z = \frac{1}{2}e^{-\frac{\pi i}{3}} is correct.
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    Re: The Complex Exponential And Regions in the Complex Plane

    is part B correct?
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    Re: The Complex Exponential And Regions in the Complex Plane

    Quote Originally Posted by andrew2322 View Post
    is part B correct?
    You tell me. \displaystyle z = \frac{1}{2}e^{-\frac{\pi i}{3}}, so what is \displaystyle z^{-9}?
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    Re: The Complex Exponential And Regions in the Complex Plane

    2^9 e ^3pi?
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    Re: The Complex Exponential And Regions in the Complex Plane

    which is equivalent to 2^9 e^pi which is 2^9 (cos(pi) +isin(pi)) which is -1 * 2^9?
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    Re: The Complex Exponential And Regions in the Complex Plane

    Quote Originally Posted by andrew2322 View Post
    2^9 e ^3pi?
    Correct, but don't forget the 'i':
    \left(\frac{1}{2}e^{-\frac{\pi i}{3}}\right)^{-9}=2^9e^{3\pi i}
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    Re: The Complex Exponential And Regions in the Complex Plane

    Cool, thanks for the heads up. I really have to stop doing that haha.

    Would you happen to have any clue on how to do Q1)? I reposted it because this one got quite messy and still have yet to receive a response.
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