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Math Help - Riemann surface - question

  1. #1
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    Riemann surface - question

    Hi.

    I have a little difficulty in understanding the following problem:

    Given the function:

    g(z) = z + ((z^2) - 1)^(1/2)

    Let f_0 denote the branch of ((z^2) - 1)^(1/2) defined on the sheet R_0, and show that the branches g_0 and g_1 of g on the two sheets are given by the equations:

    g_0(z) = 1/g_1(z) = z + f_0(z)


    OK. I see that it makes sense that g_0(z) = z + f_0(z). However, I don't quite see how it is also true that g_0(z) = 1/g_1(z). Any tips/explanations for why this is true will be greatly apprciated! I am quite stuck on this problem!
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  2. #2
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    The two branches are given by g_0(z)= z+ (z^2- 1)^{1/2} and g_1(z)= z- (z^2- 1)^{1/2}, taking the + and - values of the square root.

    1/g_1(z)= 1/(z- (z^2- 1)^{1/2}). Rationalize the denominator by multiplying numerator and denominator by z+ (z^2- 1)^{1/2}.
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  3. #3
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    Ah, I see it now! Thank you so much! I converted the expression to polar coordinates (which I normally do for all roots of complex numbers), and this is what made it somewhat difficult for me to see this rather obvious answer.
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