1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 225 The preceding proof is more general than that indicated for Theorem 5.4 in the sense that it holds in C ...

226 Chapter^5 Finally, when c = 0, a= d, b = 0, equation (5.6-1) is satisfied identically. In this case the transformation reduc ...

Elementary Functions Combining (5.7-3) with (5.7-2), we get μ - Z1 = M2 z - Z1 μ-Z2 Z - Z2 227 Similarly, after n applications o ...

228 Chapter 5 w-z1 =M(z-z1) where M = a/d = a^2 • It is an easy matter to check that this multiplier again satisfies (5. 7-5). I ...

Elementary Functions 229 and Hf:. 0, and the transformation can be written as in (5.7-2). Letting W-Z1 W=S(w)= --, w-z2 Z=S(z)= ...

230 Chapter^5 (a} (b) Fig. 5.6 Since a circle about the origin has equation IZI = r, the corresponding circle C 2 has the equati ...

Elementary Functions we have 1 M+ M >2 and it follows from (5.7-5) that a+ d must be real, and la+dl >2 231 (5.8-3) That t ...

232 Chapter 5 mapped into themselves. The points z 1 and z 2 are inverses of each other with respect to any one of those orthogo ...

Elementary Functions 233 (c) M = Rei^6 (R-=/= 1,0-=/= 2k1r). In this subcase we have W = Rei^6 z, and the transformation can be ...

234 Chapter^5 where M = a^2 • The substitutions Z = z - z 1 , W = w - w 1 brings it to the form W=MZ so (5.8-6) represents a hom ...

Elementary Functions 235 Fig. 5.9 Oc. Figure 5.10 represents (assuming that z 1 = 0 for simplicity) the two corresponding famili ...

236 Chapter 5 of the types called hyperbolic, elliptic, loxodromic, or parabolic, then the binomial a+ d necessarily satisfies c ...

Elementary Functions 237 and (x1,x2,xa,z) = (w1,w2,wa,w*) Since (x1,x 2 ,x 3 ,z) = (x 1 ,x 2 ,x 3 ,z), equation (5.9-1) holds. C ...

238 Chapter 5 The symmetry principle may be applied to solve the problem of finding a bilinear transformation that maps a circle ...

Elementary Functions 239 Fig. 5.11 q Fig. 5.12 rotation 0, and forming an angle^1 / 2181 (Fig. 5.12), z being on the bisector of ...

240 Chapter^5 a bilinear transformation is equiva:lent to the product of an even number of symmetries. 5.10 ORIENTATION OF A Cff ...

Elementary Functions 241 not contain z3), then 0 < Tz < 1; and if z lies between z 2 and z 3 (in the arc that does not con ...

242 Chapter^5 the heading of projective geometry. This made Cayley claim, with some exaggeration, that "projective geometry is a ...

Elementary Functions 243 these are the lines parallel to C, namely, C 1 and C 2 in Fig. 5.15, C 1 being the parallel to the righ ...

244 Chapter^5 Let z 1 and z 2 be any points and C the "line" determined by them. This line intersects the line at infinity in tw ...