1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions Show that the vertical strip S = {z: z = x + iy, -a:::; x:::; a, 0 <a< %7r} is mapped by w = tan('1r ...

286 Chapter^5 and n distinct values of w, say w 0 , w 1 , ••• , Wn-l are obtained by taking k = 0, 1, ... , n - 1 successively. ...

Elementary Functions 287 y z Go x v y z Ha G1 H1 x u z H2 y G2 e + 4'71' x Fig. 5.27 At the exceptional points z = 0 and z = oo, ...

288 Chapter^5 By using an ingenious device introduced by B. Riemann, it is possible to consider a multiple-valued function as be ...

Elementary Functions 289 Fig. 5.29 be quickly dispelled by topological mappings of spheres, as follows: Let So and S1 be the cut ...

290 S' 0 00 00 I ----!--....\ u L U 0 0 Fig. 5.:n S' 1 Chapter 5 .Ja(Z=-z 1 ), where a is a constant and z 1 is the simple zero ...

Elementary Functions 291 00 00 -- - -1---...... 0 Fig. 5.33 z-planes (or those of the corresponding spheres) are to be made, con ...

292 Chapter^5 We may note that the bilinear transformation z' = (z-z 1 )/(z-z 2 ) will carry the point z 1 into 0 and the point ...

Elementary Fm1ctions 293 00 Fig. 5.36 00 Fig. 5.37 equivalent to the torus or anchor ring, and also topologically equivalent to ...

294 Chapter^5 Proceeding in the same manner, we see that the cases m = 2p + 1 and m = 2p + 2 (p = 0, 1, 2,: .. ) lead to the sam ...

Elementary Functions Gs------~ G4------- G3------- Gz------ G1-- '\\ Go------~ L ~· U :------- Fig. 5.39 295 Thus the edges of t ...

296 Chapter^5 ID IB I GJ I I I (^00) I I I .. I I^2 3 x I I le zo IA Fig. 5.41 0 and -oo, z goes into G 1 and returns to the pos ...

Elementary Functions 297 The value of log z corresponding to k ~ 0 is called the principal value, or the principal branch of the ...

298 y z A A B B x Fig. 5.42 v --..._ ' _ frrri -5'1Ti I +Log z + 4'1Ti I ~jLog z + 2'1Ti I Chapter 5 we need countably many repl ...

Elementary Functions 299 contrast, the Riemann surface of w = * ::fZ is closed: i.e., its branch points belong to the surface. 5 ...

300 Chapter 5 5.25 Tiffi INVERSE OF THE CIRCULAR AND HYPERBOLIC FUNCTIONS Given any z E C there is a set of values of w satisfyi ...

Elementary Functions 301 w = arcsinz = 1/ 2 7r =i= iLog(z + ~) + 2h (5.25-3) To choose a principal branch for arc sin z we selec ...

302 Chapter 5 Similarly, we write w = cosh-^1 z to denote any value of w such that cosh w == z. This equation is equivalent to c ...

Elementary Functions 303 -37r/2 37r/2 57r/2 77r/2 Fig. 5.44 with G;+i on top of Gil it is clear from (5.25-3) that over z = +1, ...

304 Chapter5 Gi to another Gp by crossing the cut (-oo,-1) or (+1,+oo), or both, in a suitable direction a certain number of tim ...