Mathematics

Complete Intersections. Proc. conf. Acireale (Catania), 1983 by S. Greco, R. Strano

By S. Greco, R. Strano

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Extra info for Complete Intersections. Proc. conf. Acireale (Catania), 1983

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This is equivalent to z = x+iy. 2) arge Z = y. If z is real (y = 0), the definition of eZ agrees with the usual exponential function eX. If z is imaginary (x = 0), the definition agrees with the definition of ei9 given in Section 2. 20 I The Complex Plane and Elementary Functions A fundamental property of the exponential function is that it is periodic. The complex number,x is a period of the function J(z) if J(z +,x) = J(z) for all z for which J(z) and J(z + ,x) are defined. The function J(z) is periodic if it has a nonzero period.

This identity is to be understood as a set identity, in the sense that w satisfies sin w = z if and only if w is one of the values of -i log (iz ± v'f=Z2). To obtain a genuine function, we must restrict the domain and specify the branch. One way to do this is to draw two branch cuts, from -00 to -1 and from + 1 to +00 along the real axis, and to specify the branch of JI=Z2 that is positive on the interval (-1,1). With this branch of J1- z2, we obtain a continuous branch -iLog (iz + v'f=Z2) of sin- 1 z.

Each w E C\{O} corresponds to exactly two points on the surface. The function f(w) on the surface represents the multivalued function y'w in the sense that the values of y'w are precisely the values assumed by f(w) at the points of the surface lying over w. The surface we have constructed is called the Riemann surface of y'w. The surface is essentially a sphere with two punctures corresponding to 0 and 00. One way to see this is to note that the function f(w) maps the surface one-to-one onto the z-plane punctured at O.

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