计算积分[tex=11.643x2.286]hPE2Kn7QJGYMQpsn/p9akl1SSgNZlAaWJGWjtzKuvZst2MLlP44KyL6M58O+Dyrk[/tex]其中[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex] 为圆周 [tex=2.357x1.357]zARxh/lCxRk434m89Jde4w==[/tex] .

2022-07-23

计算积分[tex=11.643x2.286]hPE2Kn7QJGYMQpsn/p9akl1SSgNZlAaWJGWjtzKuvZst2MLlP44KyL6M58O+Dyrk[/tex]其中[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex] 为圆周 [tex=2.357x1.357]zARxh/lCxRk434m89Jde4w==[/tex] .

答案：

解 被积函数在扩充的复平面共有四个奇点 [tex=4.786x1.214]8mgBMcMVuCsbxNUYwDuWecviSdaFRXo2bIuQcmatW2c=[/tex] . 有[tex=18.214x3.071]rZM5/OPAdr7aX+kNl9iwpIf+w0dkTZfmI0tjq7+kV9wwuwc3VTCwbnOmM/0sBEuftuJ8yuhQB2ylnS5BhIL3NCg1AcO2C9cWsVdxA06/FTH3y/ZOYauWkMPry+A0rz4B81MAYWLVnomdSGxCajqX492stCG5FlmSYhwHrUBLqMsWrbmpB52JCTa3IOjB4YGuCGU9eLg4VLOr+SQEVmIS1n1cL2bN734SKCB3Bj9EBak=[/tex]又在 [tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex] 内只有 [tex=1.786x1.357]GYJ7X7XJijqizBuSGMrl3g==[/tex] 的两个奇点 [tex=3.0x1.214]G9iRwR5hEuaW1SkP9Jymdw==[/tex] 所以根据留数定理及上式得[tex=26.071x5.071]rZM5/OPAdr7aX+kNl9iwpMmPezcys5zBDj4WcsFULk8i2aO62f+S3+VCxXh/h+doZcMkXPFpK255NNWmq10Fyy2lFxS1C5weGT57TDwr3XqmV6pEydvwzMj05EhwJuB/2O3mb6h7SWxqUGUPbM1p07/t2jS8MyN6hytcZ+ptpfU9BbKBuL3v70gFGitf0mRVOoiRqwR3FR0HcHn11GrrGdUratveTm1zarXaIJLo2TNi/hSiErVqbMOsO9xHjYWYZTCQX305fj/JCrb0Mfs2mDk7zwc1vzS2Njf4VnTK2KbHpAUbNMcaRgyEPT5c/NaD[/tex]其中[tex=17.571x6.357]rZM5/OPAdr7aX+kNl9iwpIEIDeczosHkrJJpGwd3BnoDCwWm9aTBaxCU64K18XBj6YY704aYSbUK82cCUOprjHD1LAvKZS8pvmGWS6HfzTTOhHwa1Zi5jVq0JfcAC7ToHn0p0Fxyki5++oi1XKcva0pXy/hoi+rRj4fbCnGdQWLHy8creLZC/V1h6uIWDrWD/8kiqC5Fnazfolp/TejwaQ479ugFcn0isOiAeOLFBL1z6avKAhigjo4E88bo65qtfOkg9QCaiu0roGi855NTA8vAOTcuHkpU0N8EZGGaZpHQsNbmarpwT9l1gcOpQg1tUdbI95TNhISqViiHTNAd9w==[/tex]而[tex=17.571x5.429]uP8Mt6k0j65mSoCwtmQvLEd/S3WMdgO+yGbGsuXszg2Ph/rmZvtJh8ZkPcO0f80liyUfkvroJRbRQ0rnbGtRrgGpjO2xiDRhVgqWALXkX3VO8Q9tj+pFFfBlgceCZ0TTC1QWdb/Nz0+FPHGYh05iiORBf8rSeIn65AEPdDIpn9u1Mm/0QkN4kEVPC9UtCBsWJBzy+jqe5tu7lb9z3fCUshVYoNb6xeycLapwW7Y9lkHZviVuNZu2PeS//jFS8cbSfB1ibWT5KNFylzfSf27KoKceCN7j3myRHacgDz+a+pc=[/tex]它以 [tex=2.214x1.0]MVeOYouc7e3FvU1m5bCV6w==[/tex] 为可去奇点,所以[tex=9.214x2.786]79Wd/JsaQKi3RBB3vwr8380UEXBSK4lK/HvAXp+/FQ8iQtTZ+2d0btRGqUizSfeCp3PluzAnpQM/4G9TJG4Dw2nn2jOKyAgrIRaPJEdFzSxn7BI2tsntp5wvVx0+KlFZ[/tex]于是[tex=22.357x4.929]HImcpD5oH7HlUwvgLg3zhnLJI86KwuNvOqhRD1ovL/2U/ID9UZ3F2JAQWwF9ouXvXYF9WcsWuSmnmLN1HUYYpDZqq9OXtiCHStEqM3PZ0ULuvlHmq5YX49vvXLToPWVRJj20xtni4YM++avGizvWW8tzRaOaB7yKxRfyQ0+t768gIZAzmYGkndOmzfooa7c8[/tex]

举一反三

内容

0
计算积分[tex=8.071x2.643]OLgK4vtLcwvh6iS+aVoSH47SlFSnH7fcv1ecWqv2Q48KpSBaSELR9jA5b7TbKeAP1WDQkbjiPI6sFsx9oC2z7A==[/tex] 其中[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]为正向圆周 [tex=4.214x1.357]I8OhSf6VthyLUVMZG1fj7Q==[/tex]
1
试计算积分[tex=8.071x2.643]jlatHZLwxNUZA8quHvmfm+McDgm/8+2KItDp2KyNjqYs7xpit+Pl/lAEkFKzir1SAJTqF5yUefTeg9Qf3FM8Bg==[/tex]之值，其中[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]为圆周[tex=4.214x1.357]1xVUciqsWGEy9sUNHJEJRg==[/tex]。
2
计算积分[tex=1.143x3.357]5K8hOFs7AY4osKfSdrTtpX6JSAa+fSMv9/TfQQZlWpQ=[/tex][tex=6.929x1.357]yd5JLiG0MiGL8v+sTZpXeImFfDxPLfIY41FaROiE/wi33Rho2oPei6iFb9I8LQ/w[/tex]，其中[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]为[tex=4.214x1.357]pWlt79pexaj+xc9C8nY3WQ==[/tex].
3
设 [tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex] 为正向圆周 [tex=2.857x1.357]txTWpZu5gLHix+v1qFEreQ==[/tex], 其中 [tex=7.071x1.429]jCBMPg7lBfLH5lU+mMDk8ToH/s7x0Iq62khcz1UD2rU=[/tex] 为自然数，计算积分[p=align:center][tex=6.286x2.714]O5pOc82995a2SUHbLq31Rso5mNJ8dS2cSLM0atRTofcddry1LLpjXdXm4uwCgwluiU4FKdi655ox9kJE9VW1PzqC1zJxkNn/1Xp97aVuBAc=[/tex]
4
计算积分 [tex=3.5x2.643]wbDo/nS6ZVY5fjznJuTgRQpZ8xPvk2PAUKhKpiXr2I1zcNiQSecab8yhmOrAb9wh[/tex] , 其中 [tex=0.714x1.0]0x4Su7TCbst8/jAFAM9VFQ==[/tex] 为正向圆周 [tex=2.357x1.357]tzNGPi9Zx2v+1OAzo9s1Yg==[/tex], [tex=0.643x0.786]7a/0FnEfoPTV4M3l6DsKxg==[/tex] 为整数。