On Saturday, December 6, 2014 10:01:38 PM UTC+2, Aaron Meurer wrote: > > Something that I'm not sure about with representing functions as > multivalued is, how do you represent arbitrary Riemann surfaces. >
A practical way of representing a Riemann surface is by means of coordinate functions analytic (or meromorphic) on the.surface. Usually two functions suffice. These functions are algebraically or analytically related. A point p of the surface is represented by the pair of (extended) complex values (x,y) of the functions at p. The Riemann surface is then the set of all such pairs. In other words, it is essentially the graph of the relation between the two coordinate functions. (Sometimes additional functions are needed to resolve singular points.) For example, the Riemann surface of sqrt(1 - x^2) consists of all pairs (x,y) satisfying x^2 + y^2 = 1. Similarly, the Riemann surface of log is the set of pairs (x,y) such that x = exp(y). (It can be identified with the set of polar numbers via the mapping (r,theta) -> (r*exp(i*theta), log(r) + i*theta).) > > Another question is computational. How do you compute the surfaces in > general (say even for a limited class of expressions, like algebraic > functions), The Riemann surface of an algebraic function is obtained as follows. First, one computes the minimal polynomial p(y) of the function over the field C(x) of complex rational functions. After multiplying p by the lcm of the denominators of its coefficients one gets a complex polynomial P(x,y) of two variables. The Riemann surface is then the set of pairs satisfying P(x,y) = 0. (There may be some singular points. They can be found by taking the resultant with respect to y of P and its derivative P_y, if necessary.) > and how do you make cuts in a consistent manner? > Branch cuts should not be needed as the functions are single-valued on the Riemann surface. Kalevi Suominen -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/fbc689d5-4787-49c3-aa07-1886b314b0a6%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
