1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 365 the mapping when f is analytic in R (with f' -:/= 0), a small figure in R and the corresponding image in f(R ...

366 Chapter^6 y v ,z ~ lg I 0 u Fig. 6.10 radius of the circle is zero, or (2) the center of the circle is at the origin. In the ...

Differentiation 367 the conic EX^2 + 2F XY + GY^2 = 1 (6.16-2) Since EG - F^2 = (uxVy - uyvx)^2 = J2, this conic is an ellipse i ...

368 -2 Fig. 6.11 Fig. 6.12 +2 -2 +2 / / +2 +2 Chapter 6 ...

Differentiation = ( G - E) sin 20 + 2F cos 20 and d2. d0 2 (p^2 ) = 2( G - E) cos 20 - 4F sin 20 = h( 0) Equating the first deri ...

370 Chapter^6 Therefore, the directions 81 and 83 will correspond to maxima (minima) of p, and the directions 02 and 04 to minim ...

Differentiation 371 J>O J<O J = 0 Fig. 6.14 By considering the Kasner circle of f at z it is easy to see that the maximum ...

372 Chapter^6 We distinguish the following cases: Case I. If J > 0, the maximum and minimum values of ~b occur when J = lf~(z ...

Differentiation 373 or (^0 1) = 1/ 2 (A rgfz -Arg f z ) -^1 2 Arccos lf:zl lfzl +^1 27r Similarly, 1 1 lf:zl 1 02 = 2"(Argfz-Arg ...

374 Chapter6 Now, using (6.16-13) we get, for J > O, d7/J J max( dB)= min lf8(z)l^2 If z 12 - lfz 12 l = (lfzl - lfa:i)^2 - ...

Differentiation 375 undefined. However, as the point P representing f 0 (z) for Bf:. B 0 +k7r moves clockwise toward 0 along the ...

376 Chapter 6 Thus, with definitions (6.17-7) and (6.17-8) the equation (6.17-6) holds without exception. Note that %(Argfz + Ar ...

·Differentiation 377 Fig. 6.19 As in case I, the maximum and minimum values of the ratio d,,P /dB are attained when tan2B = 2F/( ...

378 Chapter 6 angle is preserved under the transformation. For mappings between planes a simple proof will be given by using the ...

Differentiation 379 so that and Hence Thus in either case the mapping is directly isogonal. We note that in the first case ( J & ...

380 Chapter^6 g 1 : (x,y,B) -t (~,77), a lineal element-to-point transformation. By a lineal element is meant a point with an a ...

Differentiation M i/-- 1 I v, I I lr-------N 1 I DI I -yy 0 1 P 10 -u,-- 4 J ---Vy----__.,j Fig. 6.21 381 The points A, B, ...

382 Chapter^6 Theorem 6.28 Let f be defined in some open set G C C and suppose that: 1. f E 1J(G). J1(z) ":/:- 0 everywhere G. ...

Differentiation 383 Thus lf(z)I does not attain a maximum at any point z E G. Theorem 6.29 Let f be defined on the closure of so ...

384 Chapter^6 Theorem 6.31 Let f be defined on the closure of some open and bounded set G C C, and suppose that: 1. f E 1J(G). ...