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metgt4
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Homework Statement
The function
f(z) = (1-z2)1/2
of the complex variable z is defined to be real and positive on the real axis in the range -1 < x < 1. Using cuts running along the real axis for 1 < x < infinity and -infinity < x < -1, show how f(z) is made single-valued and evaluate it on the upper and lower sides of both cuts.
The Attempt at a Solution
I started by expressing f(z) in terms of complex exponentials:
f(z) = (1-z2)1/2
= (e-i2pi - r1r2ei2(theta1 + theta2))1/2
= (r1r2)1/2ei(theta1 + theta2 - pi)
But I don't understand how to evaluate this on both sides of the cuts. I've searched the internet quite a bit trying to find something to help explain it to me but couldn't find anything so I've come to you guys!
Thanks for any help!
Andrew