[b]解[/b]      先给出格林函数 . 与内问题的作法相仿，设 [tex=1.357x1.214]nVlWDiaq/AF5IB+/Nxj18A==[/tex]为球外一点，[tex=1.357x1.214]lwYrWg2xXGqvIYUg4lRd+Q==[/tex]为[tex=1.357x1.214]nVlWDiaq/AF5IB+/Nxj18A==[/tex]关于球面[tex=0.643x1.0]u7XUci3hWIE/S+TBToDPxA==[/tex]的反演点 . 设球的半径为[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] . 以球心为坐标原点[tex=0.786x1.0]XhVNsLJz3AkjM19LvAbO7w==[/tex]，建立球标 系，设[tex=4.357x1.286]IP7cMDbQJ8AaRMwiO75utVRfgFLAotx+KmDjr+80xZs=[/tex]，[tex=4.357x1.286]d6ZbM+mT9glgH9jNV99vdqCXpW5aHtgYK8qP6yqE44s=[/tex]，则[tex=3.714x1.429]jXOriK9dwqAZezIUjOxRrVxM3JBFLXBrPhUKh3ZMwJA=[/tex]，[tex=3.143x1.286]3vS8AmhHpYba3mgxe3dRUaV1QO63Spmuf55c9t6Ho4s=[/tex]的坐标分别为[tex=6.0x1.286]dxD8LCzKQ2fl6URQZo80jtmGeWQJcwF0Uw1kk7S3L2F4+tAotHTUzSbupFDFGO5wHGxYLf1JrBdZKdBwnkqgJw==[/tex]，[tex=7.286x2.786]XhqAThoHaVLGv/6L0OPbqtUih+Fych9W8lH8JMkHJpAAjE5tNr4BoSAOZ9oF5gy1KwjRc1PRsiQ1IDIKjDfv1vbrpnCIBxfHr3M+rLBDjf4=[/tex]，对于球面上任一点 [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]，有[tex=6.643x2.214]P9Zc1vkrPSGdoeXgnIZTFDthRG/UUYd0/P0CgF/1uLeWkPsVXzRqlH4LYwRJxZ1+[/tex]，故[tex=17.643x8.929]I+OMvqIcMMjz48G8xx2pwpSbWBQ/j5u3fpNfpEOD76gMLeQu4aa51xAOBKLuTfZwGACXXRgyNEcrmtGGhTcfyHNSQJ0VxcM9i8FjGUTAL9BEetI/Ks8NaRKM+w5Z3t2JWTwbvOaxtQguparPr9uIvkMy3AJeUb2YA2kr+dkjkh5HTf7A8xOAKmW7FeyNyA+7VzHljnZF8N7Ht1rSZUJtbX8yccHNQL3Tds05SxYpEUkXEWZaeMXYIfHYwGlsS322aqfc2ypzBLUww1iKMTrECYqQa+RwTkCQAp0TTgw+pRgIxluDvI0G/eG4G9PNeuQfzIBMeSUD8NNkr0YNStY8BJsqkGamLBNQgGMEVX5tglt/syO2YV4kcWphp2mvBw5ZhhR0LDU9V1C1nWGo5iwr0OsGqp44NHErHbAcM9j92vp1L3HgEBf+N3GjhTh7iC3U[/tex]其中[tex=3.643x1.286]BN9QErzcBU2cxPACf6Hs8Q==[/tex]，[tex=0.5x1.286]h43hk9rvfl6MMCCLibYZ7g==[/tex]是[tex=1.857x1.286]4u7hZDBPbr7rcIYDL8gHwA==[/tex]与[tex=2.143x1.286]V1YIF0mmrvObfZuctWJobg==[/tex]的夹角 . 再求外问题泊松公式 . 若再无穷远处[tex=8.0x2.357]+ZWNUza2e/yTQE+wJglZOiwUgpt0iIy1FnUWKVWfZZP8ALY7sDT45Jj3bDplnnbX[/tex]，[tex=7.071x2.357]V9fVXReHUrcmKJSTnoNlSyaQmlhMyStJlrBtVtW2OlfShig40zQ3TjV+kACOezuJ8YQKv6MLKAsCjP9iIZjKZM4zph8wSw5aC6uSyXcW+WQ=[/tex][tex=4.643x1.286]Fj8XZLEQij2Ld8kiz5KxHavSFXoqebBVScH1MT6WJd7x90f8ezdn/WghKR3WjWEI[/tex]，则公式[tex=3.857x1.286]X2TVVaXtAn2o0Kxzw65SWTUmvNxcTEJSZKPHcb4VMn4=[/tex][tex=13.071x2.857]hvx+DhEH3TiAQdFWjWPCPEX1kw6xDE/WEiHMGp/guOXvhCrVlk3uhKzwinrntBUoGFO0zerfSlk7lIOg1xjz2/QsiTSjhd9nE0jpAO4xLvJ86CleQyQSAwkEDSths0Z8q9Xvp7X5CBS4y+xTPcbdj38qWrT7TeOWQKqJ2k1Lz+Vag0wVMXLsNvLL1tnvrss4[/tex][tex=5.786x2.357]AkdX+LOrkPjrvEuILMMsnCtMzGLLq+IBWSpyfOnqjBOtBqoIWtO/Xrnw9N8l2m5/GayBgCNp8WHYDEBFv1gDZQUwAT8Y+LFENhJvVVB/SM8=[/tex]，仍成立 . 其中[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]指向[tex=0.643x1.0]u7XUci3hWIE/S+TBToDPxA==[/tex]的内部，故与内问题解的积分表达式的推导完全一致，利用格林函数得[tex=3.857x1.286]X2TVVaXtAn2o0Kxzw65SWTUmvNxcTEJSZKPHcb4VMn4=[/tex][tex=11.0x2.857]C9d3U9O3UZeiP/6VKALYNcvu+hLRARuHMprTvuc70dTHuz1TDnUBNVkzL6LO6CHcNYoCQYotaRKbnUM0ZuKKPCJLOAXqSVhpyHMKSmyUOmiZd7RZwgTpltoQlZ34bBhFs1Y5FU0cb3Yo5OfsreYe0w==[/tex]，其中[tex=0.5x1.214]gNOHIx2AGu3qP//Yn7oxrg==[/tex]为边值，即[tex=2.786x1.357]F0V2BsQ2nGVLoKWgX8pYtW1LNNTZgTdGNwnWRdH3Ofs=[/tex]，且[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]指向球内 . 因此与内问题推导完全一样，可求出[tex=1.571x2.429]aoAtmkWSHYklGULM9bBrEjWkD5Uo+ogbzqgkcLUbUY4=[/tex]，只是法线方向相反，结果差一个“-”（负）号，于是得[tex=4.643x2.571]fjjoNvT9SpOcAeqWdjfzYrBuWA0QV4skDc0HHYkazmoXm277LRf1/Wzn+QVPR2dCSHo6mCgw55oXRVziyrkMZw==[/tex][tex=5.0x2.429]3ZWkpJ2ZpwxjMvbC78eO7xdkS2LLncnYYIQNHMbJZRTAgVzcEQma2hFyN+0xduLnQf/OmErxZhrr4LnPNzHVog==[/tex][tex=13.5x2.929]39Q2rDnc13EinGeyM1NjKVLIf1ymjh91JYgtbtEk78jZ/9uT+klm4rosj8LXIUGxKGehr8fm2x2Vb1X48oaXYJ1fFTdVahfDACC0KvBJyex8Uum/GCgnrLtbm8pFywSacv4GZidQ2bPiuW2f8FNKanWCj6ibwr3mvjTZ6Y9tXDuMzojAgEI8rxFcDRsj+XHU[/tex][tex=13.643x3.143]zNC+R2SI+9UJOiidR/waquYihy9loEwmXd2OMiezxidgTRijUFkTCYjqSKvOKVqxnTbjyJ0LJQANL6NCSQWL3LPCHfltCbrCJDFDYnJDfkkWPw6UucvyOxoXLHP/8kf24GpnaPfcfsS+Al5RJQxOYTk39m6BXsOHHjDzmwdN2Or3JqJjrgvCjZZblYAb5KKG[/tex]，又[tex=3.571x2.357]X6QnofO8jHW7wUBc73mYLJjQescybvQflt1mEA54zHmgYqX2RCt64HHqcpTCHLaA[/tex]，化简得[tex=4.643x2.571]fjjoNvT9SpOcAeqWdjfzYrBuWA0QV4skDc0HHYkazmoXm277LRf1/Wzn+QVPR2dCSHo6mCgw55oXRVziyrkMZw==[/tex][tex=13.0x2.643]39Q2rDnc13EinGeyM1NjKTRxiPniSfHC4fJYhuuiZstdfhHozhvnM7O19WhBS3B7yKB4Oo19WrP5MWCHN2MY+hFv1sCEHYHyobQfMPewVM5DyHXCYTwx/E/Hg5SoZrHrLatbaqc7e9dA96iW7YS2Sw==[/tex]，所以[tex=3.857x1.286]X2TVVaXtAn2o0Kxzw65SWTUmvNxcTEJSZKPHcb4VMn4=[/tex][tex=18.786x2.929]hvx+DhEH3TiAQdFWjWPCPD4aPfHWrBWrJpadADgbmGJdScMZ5DClerSY4KjVk9VDZ9qEOEnNK1NYCQ2ZT/gBztM4bFLMrx0Zyly933Q7VASWxk7R5IRpZio/c7FoIPXC/hE5l3RkdkU8DdImGMfFdBKLYc/GtuoECFKDtkSu2HrdWPyENeH47Vc2OnPHOkcsSlvgXaT4lORVOWZpCFk3bA==[/tex]，即[tex=3.857x1.286]X2TVVaXtAn2o0Kxzw65SWTUmvNxcTEJSZKPHcb4VMn4=[/tex][tex=10.357x2.429]reahECGiyH/pJ5uodJsJFfbf6MS7oVvTpI9oL8/dvGfaAKBF2AzF+87BsfuS9pz65/KZXtcYKa9ER8JZZAtST8WQPOpIIzraPjBTDicKzDM=[/tex][tex=15.143x2.643]652O8U17xGi+6pNHqCO6PjEVfxgpsqrVtyhqzPWpx98dWS+Z3s1kQ/Vay5SFcnDoCA5CloXJBJEQFHA8R5tiKkfIj3ct8mZ3Ixe91YUCreIXnXqrPLdIxRETof22sPO6f8873OY71Wo3/ymy4nmR1iOsBd18B4QqOrsV5Aqaprs=[/tex]，其中[tex=8.5x1.286]XsTNVE+h+yxAgDbcxC6ovdQn10rMFw7RRqzJiF/mCNY5JVX8lSSH7mzcHP7dnDU4[/tex][tex=9.5x1.286]RmnkBQ55yvZby1GG+GnP5P8uHfb3mPBu+zpbu6VJcXHqbpfsva/m9uxlqlQmuEkDtO+bHwSnZw1+qOBaigQlLfxuOOqYtcyi/XCWdFocA9c=[/tex]，因[tex=11.857x1.5]Oczr761pxvhspcZF9Ve+ov2Og3g3BSWEUTxAX4wnqlNs2+cuVZZ4FyiL+sFn/RlNy3P6hjzWBs1Xiuqd6zRnqzSZ+F3fpoiHK0c6nFpRcVQ=[/tex][tex=7.214x2.0]NhAxzF8HnwvAFoDfcDZX1zySK2tx8z5QveWieY/C/jHhwWBfvjV+XPPUVju7RNQp9M9XfZ+3wqePxCeR8Waz4A==[/tex][tex=4.071x1.286]OdVKWrLapF/q9OFIR98HhZTnnSD7LitMDgeJ1GkJDhuEq6x9o8b2PuFNhyU3mPEr[/tex]，所以[tex=6.357x1.286]7OK1EKxBdWwah8IfvdjNB8RPpBbtDiwieqV2zEfNmce9m8l4xEJC4rUGbpZZxJJF2gVBpZ2wOg+lNLktZj4IYhcEnrK1jsz2QW/LRrxKp/c=[/tex][tex=7.0x2.429]Er0tfzNBgBUFa8yWnQN8YMOW+9ZVc7gU0C8VAx/NfOXN7lhFaexN0lvzIgLvowRqe7nynBzte5ChaRUcsM6QVZp6N6RaGmp7ugELCufJuvg=[/tex][tex=9.214x2.429]43DHJW5hNMMx+YjFVGcc+n22g3NS48jaUzRMpMUx32tnBcKqfHRuu4xOjXfnI7OFuEq+QjVmKj7RjrqAySAbtu956yUAVDE2e/DdNWMQppM=[/tex][tex=6.714x2.429]3ncmvIqRc7vYvry9nIg17KvYsikYiPOURYesb4YG2HVQe2PC3VJPMOssnoUgRNz70vOgeN0GAjLw/fewazEeuQ==[/tex]，其中取[tex=1.071x1.286]/vZEgalrrOYkhzS9SMg+fg==[/tex]，使[tex=7.0x1.286]PJE5LGouj8KoI/JoHWhAtraOJ43IM1neEQyflDbCC9eim8pMQDF8uRQAiEffHVrb[/tex]，由此知，当[tex=3.429x1.286]O5eTiOH5f+mGhytyb9JSgPodfYxk+RYs4sNNEBhDlvlLFkgDBuEEMUEIK8CRMNeH[/tex]时[tex=4.929x2.357]Wt+ZipNXidEuT+zPhQiXuNGop11DSZHc6YBRX6luymwR11Hs4+CwjJGqvJkb2uMA[/tex]，同理，对[tex=0.929x1.286]9jyPYU8qlsK6BUzWcnoBPQ==[/tex]求导后，可验证[tex=4.929x2.357]Wt+ZipNXidEuT+zPhQiXuNGop11DSZHc6YBRX6luymwR11Hs4+CwjJGqvJkb2uMA[/tex]，即以上在无穷处所要求的条件是满足的 .