I am a mathematical hobbyist teaching myself complex analysis from Gamelin's book. I have already struck a problem. Would appreciate help.
On page 16 of Gamelin, "Complex Analysis" we find the following text:
"Now we turn to the problem of finding an inverse function for w = . Every point w not equal to 0 is hit by exactly two values of z, the two square roots . In order to define an inverse function, we must restrict the domain in the z-plane so the values of w are hit by only one z. There are many ways of doing this, and we proceed rather arbitrarily as follows.
Note that as the rays sweep out the open right half of the z-plane, with the angle of the ray increasing from to , the image rays under w = sweep out the entire w-plane except for the negative axis, with the angle of the ray increasing from to . This leads us to draw a slit, or branch cut, in the w-plane along the negative axis from to 0 and to define the inverse function on the slit plane C\( ,0]."
Now Gamelin does say this is somewhat arbitrary - but he does not motivate this "branch cut" idea well. Why are we doing this - presumably because when arg(z) = it repeats the same values as when arg(z) = - and so we are getting rid of repeated values to achieve a single valued inverse function. Is this the reason for doing this?
Presumably we could cut the plane at another ray? Is this right?
Another question is why are we removing the point 0 - the function and the inverse seem to be defined there and there is no question of repeated or multiple values. Is the reason something I will discover [possibly related to continuity or differentiability?] later in the book