试证[tex=8.786x2.357]TnUDpRqmp498lq23BsppU5DHMFH1fKl7zMotzn8+SyM2n30HbyVGiZLMXCNDIzQC[/tex]都是调和函数,但 [tex=2.214x1.143]hSY9o2OT3oZRiVc2tzqZ/w==[/tex]不是解析函数.

2022-06-19

试证[tex=8.786x2.357]TnUDpRqmp498lq23BsppU5DHMFH1fKl7zMotzn8+SyM2n30HbyVGiZLMXCNDIzQC[/tex]都是调和函数,但 [tex=2.214x1.143]hSY9o2OT3oZRiVc2tzqZ/w==[/tex]不是解析函数.

答案：

因 [tex=16.357x2.714]aoAtmkWSHYklGULM9bBrEq7HPnahanM5PrY8LstZJFpBy+BbLqfMuIvbg21xjytCDS5qgXEPI9ztHzXk8PW2H5QL8iIpmHeOD7qeFb6ScLOOp+0sN/z5yV8A+RZKUZBPOFdk40/cBv0AkO5hUEF9Li6oZIbm638NYv/k6E0sSChFv/7MaDjIslFU1n0UMeA6+fVG1wN52wO0TT+RwPDqbJwTwUYCyKK47nzWIXZXCmo=[/tex], 则[tex=10.286x2.714]RJakEEAq6IZY8uMTdSFcEHQciZWroG/aWxAiK+JrmCvrnoRNzq5656GfaHMXlCtHbmpycn9BD+I8y+UC2GRal0K6YtoeJsxlqvx1zet8zKRazCoqvqf27FPw1VhT9HXj[/tex],故[tex=4.0x1.429]B2kw3mgCDd95tTJy7mKUTA==[/tex]是调和函数.又[tex=16.286x2.857]OTYtdAptVAXC+ZYqi3USUHaUR1sb1If/SgH7nqE4gB3csNl+2gG1QuWGzntTso3/P6CvFjnE34eUn2ZIVJMPSr7BCKQ5Khr0YVz6LOHU4EsDdRHslAaEEgy8ezXsW2oagZqIXU9tG4Tq9XxmirEYLpZkjq1dSv21C3nzZq4mwnc+CO48J/vcu1ZmoZWt9GeHHvKxkbXC73ulJ5lJ31N2Ir1Givs0XTRkZwxD5C1+BaM=[/tex], [tex=21.571x2.857]OTYtdAptVAXC+ZYqi3USUJcVUS85OJU+6MaZYH13AAKqRKwHjcMwPQK0YVSGFftYx2ep/4xf0gCkgTkSBAMW1dqVJhO0HqCKkEpOnvc3RDFesxm/BDVI7t3QyasPh2+6WV0B8ih2ueeMsT1K45Qw8TFf3/qfPnyX3l+3p+RfCoE2Jnt3zsfvqLvhExyJ2npuLH2jJPhQ417DPOMQuIZCHSyJuhEJjMMFbf044SVU/3ZE9UNTKiuptA6EK4BcOmUbwaYA4vyWcNaW077KvwW94vIdQ5qBHDrjTssNEAll7BhqSWzVAZMMuAoealsK+mtY[/tex],则[tex=6.0x2.714]RJakEEAq6IZY8uMTdSFcECFAqGeLy+Ib++AnMw1sjNqYYxVCkjgK6zno94ok7TyMf4PI7TMYZM8ckuLiRSgIeMMccldNNCF5n3ou4+wJA3M=[/tex], 故[tex=4.357x2.357]RIpb8PfVKHAUFW5smP9BYrAaLJKiSCzkWlksRW0d3ic=[/tex]是调和函数.但[tex=9.857x2.643]aoAtmkWSHYklGULM9bBrEiQHxo9nrv2YmMqWBPVIXjwG5mlFoUP3TC4cL243yyvrBvjMy4vyjsaEahIjm3qIOvK+diVpuKBok61c8HdFRdNA6/NsnFId8cLOtUzmFwdL8t7Qy1KO6E+jCZwR9beUVZ5NWVHXi1+rONDJvYF+06UEMAFmZ+LBYQ7Dt5hQ6MiH[/tex], 故[tex=2.214x1.143]PBPTc6BIXn2Do4wgAEsvvA==[/tex]不是解析函数.

举一反三

内容

0
试证下列各函数为调和函数,并求出相应的解析函数[tex=4.143x1.357]LogYAzAvCq1eGBWwADRiTDEXgIiuKOaEt6GvKoBkivE=[/tex]. [tex=2.357x1.0]JjGYng9ZYe1dDhs7Z0+1Ww==[/tex];
1
证明[tex=8.643x2.357]TjeTSTmrfxPUDs+Gddz56KhzWukytBbGEKdpJPh+M4SNgOjNNYBrgNcDNHNrb0NE[/tex]都是调和函数，但是[tex=2.143x1.143]HFrWobbamEny29cerhsNRw==[/tex]不是解析函数。
2
设函数[tex=1.786x1.357]GYJ7X7XJijqizBuSGMrl3g==[/tex]在点[tex=0.571x0.786]HXNXn3AXpwdIpZt8+6oCEw==[/tex]解析，试证：(1)函数[tex=12.929x4.5]q2S0D0+eu5is3kG/mEUjHSlNB1bZoFlUTgWeVFDXvs06CW5UcDwYb2/+XahQG1kodiyblpTMvl/G3aW6dwKqZ/FFdvdZYC3cGrj8wGMcTh5OUY27gwkCMhA1lM8RuEw7dm5H3ul0pIGgsDknUCN43rwTTYqIWYXtWCDzdYu7QAM=[/tex]在点[tex=0.571x0.786]HXNXn3AXpwdIpZt8+6oCEw==[/tex]也解析；(2)函数[tex=12.857x4.357]xpGDzxuVmPETrl3cOmxwj4CSvDYDMj2vKWj/lrsyP3VnzctzHAug3OfZH8sWN47JjYVfSgDer22UeZMJF1t0JYJk/gq1v9JWh8uBwrVutbDPCjhvUILVOMI8cIbgJByo8MNwvqO87w7UDx/zRbnV/A==[/tex]也是一个整函数。
3
如果[tex=4.714x1.357]k7ZZy29fAPTldYCnWZx7/A==[/tex]为解析函数,试证 [tex=1.357x1.071]og+VaJoemW8SvHvpKGJoig==[/tex]是[tex=0.5x0.786]pmD1JbahT9zMRAbBNi045A==[/tex]的共轭调和函数.
4
试证下列各函数为调和函数,并求出相应的解析函数[tex=4.143x1.357]LogYAzAvCq1eGBWwADRiTDEXgIiuKOaEt6GvKoBkivE=[/tex]. [tex=2.714x1.0]EU7J1uVpibU3m61hnB5u2w==[/tex][br][/br]