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    I've been looking at a question in which we consider the i base logarithm of a complex number, i being the imaginary unit. I was meant to find an expression for logi(z), and make a plot on the Argand Diagram.

    The expression I had no trouble with.
    logi(z) = ln(z)/ln(i) = (ln(r)+iθ)/ln(exp(iπ/2) = (2/π)(θ-iln(r)).

    The Argand plot gets me.
    Treating logi(z) as a function with θ and ln(r) in the place of x and y, respectively, one can treat logi(z) as a polar function:
    logi(z) = (2√2/π)*exp(i*arctan(-ln(r)/θ)).

    This gives a circle, with a fixed radius of 2√2/π, and an argument which is a function of both r and θ.

    The answer says that the true plot is a region where the real part is bounded by the lines Re(logi(z)) = +/- 2, and the imaginary part is defined for all values.

    I'm confused. How is the real part limited to within +/- 2? And if it was a rectangular region, wouldn't the imaginary part be always negative? The imaginary part is proportional to ln(r) by a negative constant, and ln(r) is only defined for r>0.

    But also, assuming I went wrong, where did I?
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    (Original post by Nuclear Ghost)
    I've been looking at a question in which we consider the i base logarithm of a complex number, i being the imaginary unit. I was meant to find an expression for logi(z), and make a plot on the Argand Diagram.

    The expression I had no trouble with. And if it was a rectangular region, wouldn't the imaginary part be always negative? The imaginary part is proportional to ln(r) by a negative constant, and ln(r) is only defined for r>0.

    But also, assuming I went wrong, where did I?
    Ln(r) is is negative for 0 < r < 1, so -ln r is positive for 0 < r < 1.

    Haven't read the rest yet, not home right now, sorry!
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    (Original post by Zacken)
    Ln(r) is is negative for 0 < r < 1, so -ln r is positive for 0 < r < 1.

    Haven't read the rest yet, not home right now, sorry!
    Oh, man, that's obvious. I'm dying inside...
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    (Original post by Nuclear Ghost)
    Oh, man, that's obvious. I'm dying inside...
    For the bounding of the real part, try letting r tend to infinity or such and remember than arctan (infinity) = pi/2, that might get you something worthwhile, although I don't have paper around right now.
 
 
 
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