证明函数[tex=7.214x1.5]S1ftoyHK1niA72U6OK1yRg5tEHfHxeO1QjO4ZoPI0AE=[/tex]为调和函数，求出共轭调和函数[tex=2.857x1.357]oni5YFYZg9r1D8AXbqLQGA==[/tex]与解析函数[tex=4.643x1.357]T1b4MpRp1jts8m/9pqZ81hEEUhkSW+IaVg1mIAJLtGI=[/tex]
解由[tex=16.214x1.286]UGQi9IH0y7ZLrnEJ14Af5e0AnvPgT/iihEJMEMgy0BbE/JViHBIqsrC0rMSUGzjycwgmzonb7NjM3rdrRGkmVS7Sf/ehu+QfRzCI3dvHAIM=[/tex]得到[tex=4.571x1.286]cR97ta6q7TpJB2GAA6KHYs0Jxt5E9j8M0zxjCYBpWaA=[/tex]这就证明了[tex=2.786x1.357]dal2CT/ildxtMhXRn1xYUQ==[/tex]是调和函数.下面由[tex=2.786x1.357]dal2CT/ildxtMhXRn1xYUQ==[/tex])求出解析函数[tex=2.143x1.143]zIEt+xmL5WTc6jeSqgaKtw==[/tex],这里介绍两种觶法,解一:由科希-里曼条件,得到[tex=7.786x2.929]fnpmC2J6JmQBLyo5NmGAz2ttwj0DbIJGomtaTSTvzdockS/uc8X7rgqqZFIE3xZsMuSh6QQ5Wu7lRYJSU80/AgOWe0L7AMN86eCChpULgYRYTF5B0+xH3pPlY6xrVQHH[/tex]因此[tex=19.214x2.643]Mv7KIV0euXS3QNxju1THQvJ9uGW67VmgTdbcY/J5WNi35tKGs3KMeFOLdlHEcus9BNjZNdIvTOwTeB4zf+s6gQzZge7GYCoiJmAMQo6Vctw=[/tex]这里[tex=1.929x1.357]gevdRucC7HJ1xJIdY7uaJw==[/tex]是任意实函数.为了确定中[tex=1.929x1.357]gevdRucC7HJ1xJIdY7uaJw==[/tex]将上式对y求偏导数,得到[tex=6.429x1.429]vI3dYBSb77hhgeqXnFEeXqAfTwy4DZm51x4ExE9PaEU=[/tex]于是[tex=3.429x1.429]jbFjs5UFSJ4jmzehHWC/Y4+koa3zjPSIsQOhIJyO1+4=[/tex]积分后得到[tex=4.286x1.357]7URpYD0uZK/ECHFx/Qzn9O2V3AIaEY26Fp9MeUS5Ksg=[/tex]这样就求得[tex=7.571x1.357]OcrHkRC28h/hSMjZKOUjAAL4/sI/lwq9V1c4XRC0lRQ=[/tex]于是,解析函数[tex=15.214x4.643]ExPHhRoWulIJgsblFJc6f6+C04ZC9Qy2Q8E/dDNOhcBQcTTXsbaLHHRaJyN1PP7ang0UVnWc7rtuug4/gwUQm0i5eJ2J+n5HEca1Oj1AODy6ia7rfG5qRROvL26jt2+UyQ0BlurNb07/7AbfAuaszw==[/tex][这里根据积分来确定函数,带有一个任意常数项[tex=1.929x1.0]BTIL4gokK2Bl64i4c23+Fg==[/tex]解二:当[tex=1.786x1.357]5GXDBi3fRz6I6Au55YSUHw==[/tex]为解析函数,[tex=1.786x1.357]5GXDBi3fRz6I6Au55YSUHw==[/tex]中不含[tex=0.5x0.857]bIrzjwgFBYhTQIGgX0qQZA==[/tex]即[tex=3.857x1.357]QCIV5sxWDQ5Ngu4d5UrQVxhJoHxgWTfjN2YbigyhMtE=[/tex]由[tex=4.714x1.357]AuEp2BXwmP9I0I98JrxTeQ==[/tex]得到[tex=4.714x1.571]i9MPzwGCqffwslMoXyMMixViJRrpEpOCHUgvqgKXh7g=[/tex]可以解出[tex=11.714x5.214]fnpmC2J6JmQBLyo5NmGAzypirEXRs06b0wmtGg7bG7mI4oJAnHyK5c4jUgC8DiWk1gkJs02HB/odFbLApbTl5bVjEfD9EOMbeflP3Jkg/40RtVTPYZ/sgOW/BQo63WSq8PA6edzmK8Rga1dlxkmpJrMCgB+zMdTfbGTC+N/ZrD88X6qZRCOu1DojNF1IafUx[/tex]只要把调和函数[tex=2.786x1.357]dal2CT/ildxtMhXRn1xYUQ==[/tex]写成上述形式,就能求出解析函数[tex=1.786x1.357]5GXDBi3fRz6I6Au55YSUHw==[/tex]现在把[tex=7.214x1.5]S1ftoyHK1niA72U6OK1yRg5tEHfHxeO1QjO4ZoPI0AE=[/tex]化成上述形式、利用[tex=5.929x2.357]DdWp09f8GYl8afjeqrBcaY1Kxeb3txYsh0NgZO1dmpA=[/tex]与[tex=7.0x2.357]Uo+OluqmWE1c/jAxCqgLc+Vk0tqkSBdRL4fNeHv1JLnRa0vmc6+vr7eYq+VbiMqx[/tex]得到[tex=15.714x6.357]AFFFGbgoruOM2yuH0ARvEhmBFVAr3Loitg324ZQoQGF07iO3kPvm+ZKdxBAZNwXN2IXwuSTUKE6u7MfyEeA3lbTxJgqpK/JSL82UpND2zt9dmmnFVM/jIiKTAWyVsi7i5Uc1toKXBa3iHGEWiw8tTpWTQ2a2ngbd7OunKTYZt+pPbsEr8/CnStYv7ZRK+GnfF1t7aSWqII0Nd68ZNJ+7336OQ5p5NDE2SiUW1O/h3v3o5IbN0l1AGzK9AaHF5p/7M2O8wXadfdlI4macScMGnHKoQWowtX5zsB4rfubn4SjBxOC6ik+HQfw4Obx1tlorsOajHSSVOdBnp1r1Mim+VycfgjkUkjAQCBdWfPaHTlc=[/tex][这里已把[tex=2.786x1.357]dal2CT/ildxtMhXRn1xYUQ==[/tex]化成[tex=9.857x2.571]TbcBKCYgvZRkfkw4i0Khkeii6ufz9BGUj4BCoom8LHy5+g357GBjb42W58DAfy3T[/tex]的形式。]于是,我们得到[tex=4.214x1.5]dJCwt6TlCgPbZXtY3N0FgBlqCNqxOMvjc2VTc0q3isY=[/tex][这个方法未应用积分,因而不含任意常数项[tex=1.929x1.0]BTIL4gokK2Bl64i4c23+Fg==[/tex]]
举一反三
- 证明函数[tex=7.357x1.5]6fiPBcL0iUH5inRtC3/AALZPJbKDC9g8o1un1ZPf/s8=[/tex]为调和函数，求出解析函数[tex=4.643x1.357]Vk/CVBWmjrgkz3CajpNVVg==[/tex]
- 如果[tex=4.714x1.357]WVXxmENOyLsrAK1u24ZaKpfiudWoCMFKE7yFzwDqM2w=[/tex]是一解析函数，试证：（1）[tex=2.143x1.786]4KZ/pO3sLF10383T2p50AW8R8WCm4Eix38GcIab+i3pwQwzF+fKFx+fNj5ugVqci[/tex]也是解析函数（2）-u是v的共轭调和函数。（3）[tex=8.929x2.714]VGXzV15psxV0cBMwKVrVbuMoOXdV+Yk2MpPVTJAvfxMCkIE3bPD/y0Sxfc1i847aFfKM2ml7yjnGpx5L1BLATZBsa7LoRJpwet6xLTvJptc4CCpsVuNX5Ot7Bqn2RJCiclGj4NQRw7fbKhp2F0ajXw==[/tex]=[tex=8.857x1.643]Q0Ezd43LDeNGSFF1HD/X+nVEh77NG1HPyrxZMPH3igXsS6p9dc+DNDH3M8YtOEmzDfM1ltfMQmlTNS1Y3AOXC9SS+kODOYzv283/UaXaVtQ=[/tex]
- [tex=5.143x1.357]2Js34AY0MRDQ2/Apd8rGGA==[/tex]是一个调和函数,求:[tex=2.857x1.357]oni5YFYZg9r1D8AXbqLQGA==[/tex]与[tex=2.786x1.357]dal2CT/ildxtMhXRn1xYUQ==[/tex]组成的解析函数
- 由下列条件求解析函数[tex=4.643x1.357]T1b4MpRp1jts8m/9pqZ81hEEUhkSW+IaVg1mIAJLtGI=[/tex][tex=9.071x1.357]6haMKJs64xtHBC/FnedUjWgEUhUEmTLzKRF7ViVBKDs=[/tex]
- 求下列函数的导函数：(1) [tex=5.0x2.357]X/CieCDGJ7iPQ3YFWuscHxHrcIE/dPFa9tFyiJXze8A=[/tex](2)[tex=6.643x1.714]Oj74y/L+OxY81QME5JWMcl+7PZ2FGQswwvjgVhjq1Dmb6dBU0oAjZBW7eFBVjqo6[/tex]
内容
- 0
已知函数 [tex=7.5x1.5]nFJ5eriQ3JjnYeseRMPezorIFFo1tTuptiBl91cW/b8=[/tex] 证明它是一个调和函数且求出其共轭调和函数 [tex=2.929x1.357]ffp8Fn+Z/Hdd7Z8sq55+eQ==[/tex]
- 1
设函数 [tex=5.429x2.5]P8VJNRb+PUXRSXaNQAnR3aAsyL6+p8y9A1S7KLIiwSw=[/tex] 则极限 [tex=4.286x1.857]ENxIatiC2yqgaopSQCG83qvjl0y5knXrZ2tTmOIv4gk=[/tex] A: 0 B: -1 C: 1 D: 不存在
- 2
设 [tex=5.857x1.429]grsiQIxH1QtysS2kXoDoxJ9oQQ3sGxwmnPyqBk/5AuQ=[/tex] 为调和函数,试求其共轭调和函数 [tex=2.786x1.357]GhcMUKWYfCD3K0BhvBKDbw==[/tex] 及解析函数 [tex=9.286x1.357]VXRiOJeOrIGQLgMSad8UR758hPXkWWekuSonG3su3Hk=[/tex].
- 3
设h为X上函数,证明下列两个条件等价,（1）h为一单射（2）对任意X上的函数[tex=5.429x1.214]3BrfPgAFe5dbHQTMAYnbS+118W4YAj6CiW06EKMaxNI=[/tex]蕴涵[tex=1.786x1.214]pxzkG5OdsKT9CiCwC5OvPQ==[/tex]
- 4
某人对商品x的需求函数是[tex=5.214x1.214]0m6eBd5eyK0NjuxeKfwtIw==[/tex]，[tex=4.214x1.214]I717YsPbj8Rnym1v2XQ+sFNkUl7mqUsGwbjwjXmy2xc=[/tex]，这里[tex=0.571x1.0]Za328cIB4SeR7rrzY+MM5Q==[/tex]是[tex=0.571x0.786]ZSLOI4fiO1oAbVC5M8IVkA==[/tex]的价格。如果商品x 的价格是0.5元，那么他对商品x的需求价格弹性是 未知类型：{'options': ['-10', '- 1/5', '-1/10', '\xa0- 1/3'], 'type': 102}