By Tarantello G.
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Additional resources for A quantization property for blow up solutions of singular Liouville-type equations
71828 • • •)* has only one value. Sometimes, for notational convenience, we shall write exp (z) instead of e'. 16 Functions of a Complex Variable: Theory and Technique Trigonometric and Inverse Functions For real x, the familiar Taylor series gives It is natural to define sin z for complex z by simply replacing x by z in this series. The resulting series converges inside every circle around the origin (in such situations we shall say that the radius of convergence is infinity). Similarly, define The reader may verify that all the usual identities are fulfilled—for example, sin2 z + cos2 z = 1, sin (21 + 22) = sin zi cos 22 + cos z\ sin z*, and cosh2 2 — sinh 2 2 = 1 .
Show, however, that if one enters another sheet of the Riemann surface by crossing this line, there will be a branch point at 2 = %• 13. Discuss the branch-cut and Riemann-surface situation for each of the following functions: where a is a constant with Re a = k > 0. In particular, verify that m(z) can define a function which has no branch points in Im z < k, that it can also define a function which has no branch points in Im 2 > — k, and that one linear combination of the two functions so defined is l/\/z* + a2.
Each of two points lying on the same straight line through the origin and distant r, r' from it is said to be the other's inverse in the unit circle if rrr = 1; consequently, the transformation w = l/z corresponds to an inversion in the unit circle followed by a reflection in the real axis. The more general transformation w = a/z + b would merely add a final stretching, rotation, and translation. The transformation w = l/z does not provide an image point for the origin. However, as z —> 0, w —* °°, so that it is conventional to say that the image of the origin is the point at infinity.