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Can someone here help me understand how to illustrate maps of analytically-continuous paths over algebraic functions onto their normal Riemann surfaces? For example, consider

[tex]w=\sqrt{(z-5)(z+5)}[/tex]

and it's normal Riemann surfaces which is a double covering of the complex plane onto a single Riemann sphere. Now suppose I take the figure-8 path given parametrically as

[tex]z(t)=7 \cos(t)+7 i \sin(t)\cos(t)[/tex]

which encircles both branch-points, and allow that path to traverse over the function in an analtyically-continuous manner. What then does the image of that path over the function look like when mapped onto it's normal Riemann surface?

Is there already existing sofware freely available for doing this one and higher genus surfaces? My objective in all this is to understand how to compute

[tex]\oint_{z(t)} w dz[/tex]

and other abelian integrals.

Thanks,

Jack

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# Mapping contours over normal Riemann surfaces

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