求将上半平面 [tex=3.714x1.071]Hxr+WAd0pdX8wRxoSXYGR0b0wJxfNMXBDfxtoIUjrQs=[/tex] 保形映射为圆 [tex=5.214x1.357]QxweJbstpIZh7Lp40n+TNeKDnMRS6EwVQImP9NUJeUE=[/tex] 的分式线性映射 [tex=3.286x1.357]x7PAMPfR7owPhHsb73hdeg==[/tex], 且使[tex=7.571x1.429]/LiZGsGGaupB5Hp0qPynU1WoZEIJuJX2v79sp+MxiCbKEVFwNhqsPi3AsmzlAryY[/tex].

2022-06-05

求将上半平面 [tex=3.714x1.071]Hxr+WAd0pdX8wRxoSXYGR0b0wJxfNMXBDfxtoIUjrQs=[/tex] 保形映射为圆 [tex=5.214x1.357]QxweJbstpIZh7Lp40n+TNeKDnMRS6EwVQImP9NUJeUE=[/tex] 的分式线性映射 [tex=3.286x1.357]x7PAMPfR7owPhHsb73hdeg==[/tex], 且使[tex=7.571x1.429]/LiZGsGGaupB5Hp0qPynU1WoZEIJuJX2v79sp+MxiCbKEVFwNhqsPi3AsmzlAryY[/tex].

答案：

解 令线性映射[tex=4.214x2.286]ayvftphMULS7M0JnXe8U53uRh1jRgPge1FDrJMSqEhA=[/tex]则该映射将圆 [tex=5.214x1.357]/20KVTrPrvkr0hkA3vFyhfNNaeUPI2MmwufZNFj5aQg=[/tex] 映射成单位圆 [tex=2.857x1.357]k82EtYZEdqGZt8Qufamh2w==[/tex] .然后做上半平面 [tex=3.714x1.071]Hxr+WAd0pdX8wRxoSXYGR0b0wJxfNMXBDfxtoIUjrQs=[/tex] 到单位圆 [tex=2.857x1.357]opj4T8y2mNCGRQUx+U+MCw==[/tex] 的分式线性映射,使 [tex=1.643x1.0]AY9B2w+es7mjmYNwwHo9hA==[/tex] 时 [tex=1.786x1.214]R/15sg1T4NlwAolKS91x2A==[/tex] . 此映射为[tex=4.286x2.429]jDOiDPA6vNN3NFifSHWgIbTRcHlprs0ZNrJM3EabQl9cWQauUxotU2Ma6VRDLwsl[/tex]复合上述两个映射得[tex=6.857x2.429]3npj5GOZj6pODaZZie7OWYoCWBD13nj6PCUIcVa1O+Milc56lFbPjZfhPVdz/qyOuISimRtoXJPYV0oEpXLCog==[/tex]它 [tex=0.714x1.0]RRR4SYyCqv01G5bWEEMPdw==[/tex] 将上半平面映射成圆 [tex=5.214x1.357]QxweJbstpIZh7Lp40n+TNXzzTOAQVyQWrPtpov2bpzA=[/tex], 使 [tex=1.643x1.0]eL2hKIfOTjDv29QrfykOrw==[/tex] 映射为 [tex=2.571x1.0]sCVCbTFVsuLL/0lWJNwWgQ==[/tex]( 图6. 13).再由条件 [tex=3.786x1.429]gI719JL0TEZZGtHrC3y58tdnhLpU+XFsyVNA8k3BPOY=[/tex] 及[tex=15.357x4.0]2MUOKCKWKG3blRvrC3QeFILQansI00OKvgr069XG8LJ217fVcIwNLsObuiylDvwvA3Eme4n0a2KO6syvU19QmjptpnAS/UfTzDbaK8W6HEi5iIp9EW5H7Gjyk24uDdb0flEVuRi4VQcp8N6GXKOULADheaYfMftgK2YgdNq7rHSYhfnDsJj1EAvmbVEk/cEsM1GjpdSX5IjIwsiY7BPY3g==[/tex]得[tex=14.071x2.786]QaEQoV61JU+XNix9sy5jet7sl46g9vXD/B8hdUxU/dSzyCV7+ufYc3SBk2Qs7i8vmOebXqO47XAMfpmEUHdK5OCFd/Jqlm8K/TfqIjKW26MhRLbEWZDQNBUbN28mVPeS0yj9tqQ+Rcb1TUB6i9HkrJdC4ZEZet3RHa8NtxQ7rRN9+qh+aqms/VywwklZ4ZHr[/tex]于是 [tex=10.857x2.143]Ghk0NriNTxKS/wGkuG2JyB+K5ND5u9uvLLZuxSQenMjRprbXeCIFSNW7x+2B2gvp1m0cTtFyqRNDWfvB89J5hIBXUZOSsX4QslyLGStoQ19owVFXn2Vj/CTFqHUdnudy[/tex] . 故所求的分式线性映射为[tex=6.286x2.429]K3uthaut95dTSxQJiCYF5ignbryPxqalqddWQEbIGW1syCZexrjJMH/eH7b61Cth[/tex][img=847x618]1787c6cf6c0643a.png[/img]

举一反三

内容

0
求把上半平面 [tex=3.714x1.071]gTZHdCeU5l1aCfxewscMQQ5WaHXHYdAvgh0HoTQA2CA=[/tex] 映射为单位圆 [tex=3.143x1.357]aPOOt2soTMbyWi3H3vczCA==[/tex] 的分式线性映射 [tex=3.286x1.357]AQhvwmYnVhXPQa9e6WrGoA==[/tex] , 并满足条件:[p=align:center][tex=8.214x1.429]5sppDSDoeRctVSiZSHaR9lEbaFUbnaCk4pK44BT6jTUSAxAqRF8YG8FQYLKy2idS[/tex]
1
求分式线性映射[tex=3.286x1.357]AQhvwmYnVhXPQa9e6WrGoA==[/tex],使上半平面映射为单位圆内部并满足条件[tex=7.0x1.357]C4AaF2CqxrpfMNLJcYYLVyDXZNo+dxMTE51KZ19h38A=[/tex].
2
求分式线性映射[tex=3.286x1.357]AQhvwmYnVhXPQa9e6WrGoA==[/tex],使上半平面映射为单位圆内部并满足条件[tex=8.143x1.429]UiUEh71niTJbnyfqEPJs177z+2ICPlZ4cwrXxnysM6r0KrdfyHWja8vsX+1B5CNd[/tex].
3
求分式线性映射[tex=3.286x1.357]AQhvwmYnVhXPQa9e6WrGoA==[/tex],使上半平面映射为单位圆内部并满足条件[tex=8.143x1.571]PzP9bEuqS63uxhwsiIioTCfO4i/7Q4FOYv9pI5AzvK0=[/tex].
4
求分式线性映射[tex=3.286x1.357]ySGySJBkLne3ga0KuR9uXg==[/tex] 使上半平面映射为单位圆内部并满足条件: [tex=8.714x2.143]EMxD6yYtTke281Z3UhewbHRGOX2ILPZwwgeuzYF1MycjKMm2WkwpRKthgiYmWkpRG4AhmYYjUtTaI4UWM8xcNLWEWMoa+XWdAU7jKwaNaMw=[/tex][br][/br]