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# Complex Number Logarithm

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1. 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?
2. (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!
3. (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...
4. (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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