Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 66', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Main Applications 423 Let us present some formulas that allow one to find the images of straight lines and circles for an arbitrary linear-fractional mapping . 1. The straight lines Re Xz a that do not pass through the point z -d c a - Re Xd c are taken to the circles w - Wo p where 2aac adX bcX a Wo P Wo 2a c 2 2 Re cdA I c 2. The straight lines Re Xz -Re Xd c passing through the point z -d c are taken to the straight lines ad - bc ad - bc Xa R X R 3. The circles z - Zo r that do not pass through the point z -d c r zo d c are taken to the circles w - w0 p where _ az0 b cz0 d - acr2 _ r ad - bc W czo d 2 - ld2r2 P czo d 2 - c 2r2 4. The circles z - z0 z0 d c are taken to the straight lines Re ad - bc i T2RW c czo d ad - bc 2 2 Re c az0 b ad - bc 2 c cz0 d 2 If a linear-fractional mapping takes four points z1 z2 z3 and z to points w1 w2 w3 and w respectively then the following relation holds W - W1 w2 - w3 _ z - z1 z2 - z3 w - w3 w2 - w1 z - z3 z2 - z1 Theorem. There exists a unique linear-fractional mapping of the extended z-plane onto the extended w-plane taking three arbitrary distinct points z1 z2 and z3 to three arbitrary distinct points wi W2 and w3 respectively. Theorem. Any disk of the extended z-plane can be transformed into any disk of the extended w-plane by a linear-fractional function. Example 6. A mapping of the upper half-plane onto the unit disk. Let a be the point of the upper half-plane which should be taken to the center w 0 of the disk Fig. . Then the problem is solved by the linear-fractional function ip z - a w e --------- z - a where 3 is an arbitrary real number. Changing a means rotating the disk around the center w 0. Figure . A mapping of the upper half-plane onto the unit disk. 424 Functions of Complex Variable Example 7. A mapping of the unit disk onto the upper half-plane. Let a ih be the point of the upper half-plane to which the center z 0 of the disk should be taken Fig. . Then the problem is
