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Unformatted text preview: Riemann Surfaces See Arfken & Weber section 6.7 (mapping) or [sections 106-108] for more information. 1 Definition A Riemann surface is a generalization of the complex plane to a surface of more than one sheet such that a multiple-valued function on the original complex plane has only one value at each point of the surface. Briefly, the notion of a Riemann surface is important whenever considering functions with branch cuts. Perhaps the easiest example is the Riemann surface for log z . Write z = r exp( i ), then log z = (log r ) + i . However, the in that expression is not uniquely defined it is only defined up to a multiple of 2 . As you walk around the origin of the complex plane, the function log z comes back to itself up to additions of 2 i . We can construct a cover of the complex plane on which log z is single-valued by taking a helix over the complex plane going 360 about the origin takes from you from sheet of the cover to another sheet. Put another way, every time you go through the branch cut, you go to a different sheet. We can construct such a cover as follows. (Our discussion here is verbatim from [section 106].) Consider the complex z plane, with the origin deleted, as a sheet R which is cut along the positive half of the real axis. On that sheet, let range from 0 to 2 . Let a second sheet R 1 be cut in the same way and placed in front of the sheet R . The lower edge of the slit in R is joined to the upper edge of the slit in R 1 . On R 1 ,...
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- Spring '11