• 2022-07-28
    求将上半平面[tex=4.286x1.357]Hxr+WAd0pdX8wRxoSXYGR6MH5wUJTcuJa9P72mFCl6M=[/tex]单叶映为图6.62所示区域的映射.[img=215x226]178e2da3071f044.png[/img]
  • 先考虑图6.63a中上半平面[tex=4.286x1.357]Hxr+WAd0pdX8wRxoSXYGR6MH5wUJTcuJa9P72mFCl6M=[/tex]映为6.63b中图形的映射.令[tex=11.643x1.214]dIrY2fYX7I8/IbO+oskhWkaifh5Q0RmL0PonKtaR32SDeeM37FkUPVM6QnEgqWQAkDob1GkvdziZRUe5q/12hA==[/tex].[img=372x212]178e2db15104c18.png[/img]由施瓦兹 - 克里斯托弗公式,得[tex=10.286x2.714]d4KoWWQvYjaL5X9Oz9OU03UFEGfNptqa/+TUCHvyesns9rDAMadI/jQEBW6k3MWj/SpYtrerMskzjoiGRrK1Rw==[/tex].若[tex=8.214x1.214]7MnJb8miN+STDFIRq7Q/Tw==[/tex]在[tex=1.143x1.214]LP88/si/gbw7V1lo5PhW3w==[/tex]与[tex=1.143x1.214]OZaU0ojHlAZy3SYSGWqrDA==[/tex]之间,即虚部为[tex=0.643x1.0]8+M7OwdUGZPUoOQAaQHP2A==[/tex].因[tex=6.714x2.786]1ivMIBQ1GwbtGGfLOccZ340BK9wCLqrx4Q2rWXTXX2tkXkN/a/JDSTHnWM+W15zn[/tex]取复虚数,故[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]是实数.又[tex=21.643x4.5]XXFcYjpfGz7IAusSVH3Kg6keegxcnhA8Uy7s6wqiGKbiw7IY6yf+qUeogSlvoGrELPhxGkdE7RmUKrJWy30qNdAxu2B/Hq2uwU0aIeyxESDV1mvpnA18ezWlrv8RCJDcMZvcxtChb+5qfbg8YhfYPhO/BKZGfQjPzwyMzBEBxMkboavPuHbnEqdwsi27uGBdWIJzjxrMrfV7lRzcyn1Ve/efJ0UwS2Duoat0vCyHZFJ5ORjF9kaoUgThq1ap4ltG[/tex]所以[tex=11.929x1.357]gTZHdCeU5l1aCfxewscMQdtiJmFMjUkW3rNB1v3UUOii3+DNaw35CExedpYfhqbsvkBgJqbHjSlf+92aqOAWxzhmyN+CeKC4iU1zSBLdkjc=[/tex][tex=7.0x2.429]Bm7eHdTs5Deb/D8bZZbB/99g9QN4lWp2r3tlLlbbAQHbxH9StsPEyc0HVXfiwMNH[/tex].因此[tex=13.0x2.714]1thr9HBA7AkzA+mvBx7Pbii6XRoXqxAoBjJg1e6pLDTdXYAGXV2Q6Bqt7SS9bllsp4G+h3DtbFZQiptV8wvZltzODgl0yMoA/cu7c2c/1Mc=[/tex].由对称原理,函数[tex=21.857x3.357]dD4dTJJ3HNMCSQ6EkKkXdvlYJHJswxMX9RK+TVOHWarDfrpF/v3cbegzMRZP/8vFmq+s0qJ6qv6Wu+S07lk6Kr8obiQHVOqK/e9qOs8eXW5WA1P2W58YAkArC3FdHh87ZsfY34IkTPhi25wqA9piao32vO1utkDJleVWIDXev+KWAr/ExWm7AuffavdaoNzboYmzDok9xKuIozc5qKRLsQ==[/tex]将全平面去掉负实轴后的区域双方单值地共形映射为所求区域.因此,[tex=4.929x1.571]ri17m9peEP0JJB4T5pYDAwus7FgPDsQxZ8HN1PkbEww=[/tex]将[tex=4.286x1.357]Hxr+WAd0pdX8wRxoSXYGR6MH5wUJTcuJa9P72mFCl6M=[/tex]映为所求区域.

    举一反三

    内容

    • 0

      求一单叶全纯映射,把除去线段[tex=1.929x1.286]uj8YUp05TOxtrNrRUulr5g==[/tex]的单位圆盘的外部映为上半平面.

    • 1

      已知(X,Y)的联合分布律为:[img=520x161]1803e1674fcfa1e.jpg[/img]则P(X≤0, |Y|<1)等于 A: 2/9 B: 1/3 C: 5/9 D: 5/6

    • 2

      已知(X,Y)的联合分布律为:[img=520x161]1803c332db8081b.jpg[/img]则P(X≤0, |Y|1)等于 A: 2/9 B: 1/3 C: 5/9 D: 5/6

    • 3

      求将[tex=4.286x1.357]Hxr+WAd0pdX8wRxoSXYGR6MH5wUJTcuJa9P72mFCl6M=[/tex],映到[tex=4.714x1.357]Hxr+WAd0pdX8wRxoSXYGR5miUG1s5eE2wroqedzwRCY=[/tex],且把[tex=0.5x0.786]C7x+w8+jOPZzxFrGGne6Dw==[/tex]平面上的线段[tex=3.0x1.357]hCUpMH37yix3aqPLXiFgJQ==[/tex]映为[tex=0.786x0.786]44SGfA2gQ2VZlXa1QKZD0Q==[/tex]平面上的射线[tex=3.5x1.357]14IB9GRNB+MqpAhXjIBkng==[/tex]的分式线性映射.

    • 4

      把下面各图阴影部分所示(边界为直线段或圆弧)的域  D  保形地且互为单值地映射成上半平面  G  ,求出实现各该映射的任一函数.[img=325x291]1790e20c8520848.png[/img][p=align:center][tex=10.571x1.357]z7V8adliApMHo1+fpoUtvWXX/OgbBNYjcxDstuvwZJE=[/tex]