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?