How do we figure out what is going to be the domain of the log function for complex numbers? It is easy to figure out for inverse trigonometric functions, since we can draw the graph. But how to do the same for log function?
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@TuringTest can you help?
I am terrible at complex analysis...
I also didn't get that @ you sent me; I better post that in feedback :S
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looks like all numbers ... excluding zero.
Thanks for coming!
if only I could help ....
@experimentX The function is a multiple valued function, so there must be an interval. According to my book it says the interval is -pi to +pi, my question was, how to find that range. I hope now I have clarified the thing.
I was talking about log for complex values.
Inverse Trigonometric functions have range pi to -pi, because they are (they are periodic ... period of 2pi) <--- any value in terms of pi can be expressed in terms of -pi and +pi
and of course it must be multivariable function (not a function)
Yes, that is exactly what I am searching for. For sin inverse, you can easily see from the picture what is going to be the range for principle value. And I have been taught to figure that out seeing the plot of sin inverse. But here in log z, how do I find the range for principle value. That is my question.
in a same way we say
sin(pi/2) = sin(2pi + pi/2) = sin(4pi + pi/2) = sin(6pi + pi/2) = 1
arcsin(1) = pi/2, 2pi+pi/2, 4pi + pi/2, ...
now let's check for log, the domain is going to be all comples plane except 0
ln(z) = ln(e^(ln|z| + iarg(z)) = ln|z| + i arg(z)
as long as |z| != zero, i think we will have all values of complex plane as our domain.