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**1. Homework Statement**

The function

f(z) = (1-z

^{2})

^{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.

**3. The Attempt at a Solution**

I started by expressing f(z) in terms of complex exponentials:

f(z) = (1-z

^{2})

^{1/2}

= (e

^{-i2pi}- r

_{1}r

_{2}e

^{i2(theta1 + theta2)})

^{1/2}

= (r

_{1}r

_{2})

^{1/2}e

^{i(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