{"id":756,"date":"2021-05-24T09:10:42","date_gmt":"2021-05-24T09:10:42","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=756"},"modified":"2021-07-20T13:25:42","modified_gmt":"2021-07-20T13:25:42","slug":"jixiaoqumiandefenzhidianbranched-point","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=756","title":{"rendered":"\u6781\u5c0f\u66f2\u9762\u7684\u5206\u679d\u70b9(branched point)"},"content":{"rendered":"\n<p>\u6781\u5c0f\u66f2\u9762\u7684\u5206\u679d\u70b9\u5b9a\u4e49\u4e3a\u66f2\u9762\u5728\u8fd9\u6837\u7684\u70b9\u5904\u4e0d\u80fd\u7528\u4e00\u5bf9\u4e00\u7684\u65b9\u5f0f\u53c2\u6570\u5316. \u4ece\u800c\u5728\u5206\u679d\u70b9, \u66f2\u9762\u5e76\u4e0d\u662f\u901a\u5e38\u5b9a\u4e49\u4e0b\u7684\u51e0\u4f55\u66f2\u9762, \u56e0\u4e3a\u901a\u5e38\u6211\u4eec\u8981\u6c42\u5728\u4efb\u610f\u4e00\u70b9\u7684\u90bb\u57df\u5185, \u770b\u8d77\u6765\u50cf\u4e00\u4e2a\u5b8c\u5168\u7684\u5706\u76d8.<\/p>\n\n\n\n<p>\u5e26\u6709\u5206\u679d\u70b9\u7684\u6781\u5c0f\u66f2\u9762\u662f\u5728\u7814\u7a76\u6781\u5c0f\u66f2\u9762\u7684\u7406\u8bba\u65f6\u5fc5\u987b\u8003\u8651\u7684\u5bf9\u8c61, \u56e0\u4e3a\u4e00\u5217\u6ca1\u6709\u5206\u679d\u70b9\u7684\u66f2\u9762\u7684\u6781\u9650\u5f88\u53ef\u80fd\u6709\u5206\u679d\u70b9, \u800c\u8fd9\u79cd\u6781\u5c0f\u5316\u5e8f\u5217\u7684\u65b9\u6cd5\u662f\u5bfb\u6c42\u6781\u5c0f\u66f2\u9762\u7684\u57fa\u672c\u65b9\u6cd5\u4e4b\u4e00. \u4e8b\u5b9e\u4e0a, \u5173\u4e8e\u5206\u679d\u70b9\u7684\u51fa\u73b0, \u53ef\u4ee5\u7528\u6570\u5b66\u516c\u5f0f\u5b8c\u5168\u523b\u753b. \u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f5c\u5e26\u6709\u5206\u679d\u70b9\u7684\u66f2\u9762\u7684\u56fe, \u800c\u4e14\u4fdd\u8bc1\u4efb\u4f55\u5206\u679d\u70b9\u770b\u8d77\u6765\u90fd\u548c\u8fd9\u4e9b\u56fe\u5f62\u4e4b\u4e00\u76f8\u540c.<\/p>\n\n\n\n<p> \u8fd9\u4e9b\u56fe\u5f62\u88ab\u4e24\u4e2a\u6574\u6570\u5b8c\u5168\u523b\u753b: \u9636(order)\u4ee5\u53ca\u6307\u6570(index). \u5206\u679d\u70b9\u53ef\u4ee5\u5206\u4e3a\u5185\u90e8\u5206\u679d\u70b9\u548c\u8fb9\u754c\u5206\u679d\u70b9. \u6211\u4eec\u9996\u5148\u6765\u770b\u8fb9\u754c\u5206\u679d\u70b9\u7684\u56fe\u50cf<\/p>\n\n<!--more-->\n\n\n<div style=\"width:90%;margin:auto\">\n<figure><center><br>\n<a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order2Index1.gif\"><br>\n<img decoding=\"async\" class=\"latex_fig\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order2Index1.gif\" alt=\"Order2Index1.gif\" style=\"width:80%\"><br>\n<\/a><\/center><figcaption class=\"latex_fig_caption\">\u56fe1. \u66f2\u9762\u7684\u5206\u679d\u70b9: \u96362\u6307\u65701<\/figcaption><\/figure>\n<figure><center><br>\n<a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order4Index2.gif\"><br>\n<img decoding=\"async\" class=\"latex_fig\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order4Index2.gif\" alt=\"Order4Index2.gif\" style=\"width:80%\"><br>\n<\/a><\/center> \n<figcaption class=\"latex_fig_caption\">\u56fe2. \u66f2\u9762\u7684\u5206\u679d\u70b9: \u96364\u6307\u65702<\/figcaption>\n<\/figure>\n<\/div>\n\n\n<p>\u4e0a\u9762\u4e24\u4e2a\u56fe\u4e2d\u4e0d\u5bb9\u6613\u770b\u51fa\u7684\u4e00\u4e2a\u4e8b\u5b9e\u662f: \u5355\u4f4d\u6cd5\u5411\u91cf\u53ef\u4ee5\u8fde\u7eed\u5730\u5ef6\u62d3\u5230\u5206\u679d\u70b9(\u56fe\u50cf\u7684\u4e2d\u5fc3). \u8fd9\u8868\u660e, \u66f2\u9762\u7684\u6bcf\u7247\u5728\u4e2d\u5fc3\u9644\u8fd1\u90fd\u53ef\u4ee5\u62c9\u5e73\u6210\u4e00\u4e2a\u6c34\u5e73\u7684\u66f2\u9762. \u56e0\u6b64, \u6211\u4eec\u53ea\u9700\u8003\u8651\u5206\u679d\u70b9\u9644\u8fd1\u7684\u4e00\u4e2a\u5c0f\u90bb\u57df, \u5219\u4ece\u56fe\u50cf\u4e0a\u770b, \u66f2\u9762\u53d8\u5f97\u66f4\u52a0\u7684\u5e73\u5766. \u4e0b\u56fe\u662f\u534a\u5f84\u4e3a$0.2$, \u96362\u6307\u65701\u7684\u4e00\u4e2a\u5206\u679d\u70b9, \u4ece\u56fe\u4e2d\u4f60\u53ef\u4ee5\u770b\u5230\u66f2\u9762\u7684\u6cd5\u5411\u91cf\u5728\u539f\u70b9\u9644\u8fd1\u8d8b\u4e8e\u5782\u76f4\u65b9\u5411.<\/p>\n\n\n\n<figure class=\"wp-block-image\">\n<center><a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/BoundaryOrder2Index1R2.gif\"><img decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/BoundaryOrder2Index1R2.gif\" alt=\"BoundaryOrder2Index1R2.gif\"\/><\/a><\/center>\n<figcaption>\u56fe3. \u66f2\u9762\u7684\u5206\u679d\u70b9: \u96362\u6307\u65701\u534a\u5f840.2<\/figcaption><\/figure>\n\n\n\n<p>\u8fd9\u4e9b\u5206\u679d\u70b9\u4e4b\u6240\u4ee5\u79f0\u4e3a\u8fb9\u754c\u5206\u679d\u70b9, \u662f\u56e0\u4e3a\u5b83\u4eec\u53ef\u80fd\u4f4d\u4e8e\u66f2\u9762\u7684\u8fb9\u754c. \u4e8b\u5b9e\u4e0a, \u56fe\u4e0a\u7684\u8fb9\u754c\u662f\u4e00\u6761\u76f4\u7ebf(\u4ece\u56fe\u4e0a\u4f60\u53ea\u770b\u5230\u4e86\u66f2\u9762\u7684\u8fb9\u754c\u7684\u4e00\u90e8\u5206). \u4e8b\u5b9e\u4e0a, \u56fe\u4e2d\u4e5f\u663e\u793a\u4e86$x$, $y$, $z$\u8f74\u7684\u4e00\u90e8\u5206.<\/p>\n\n\n\n<p>\u4e0b\u56fe\u662f\u5206\u679d\u70b9\u4e3a\u5185\u70b9\u7684\u60c5\u5f62, \u672c\u8d28\u4e0a, \u4e0a\u9762\u7b2c\u4e00\u5e45\u56fe\u662f\u4e0b\u56fe\u7684\u4e00\u534a.<\/p>\n\n\n\n<figure class=\"wp-block-image\">\n<center><a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/InteriorOrder2Index1.gif\"><img decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/InteriorOrder2Index1.gif\" alt=\"InteriorOrder2Index1.gif\"\/><\/a>\n<\/center>\n<figcaption>\u56fe4. \u66f2\u9762\u7684\u5185\u90e8\u5206\u679d\u70b9: \u96362\u6307\u65701<\/figcaption><\/figure>\n\n\n\n<p>\u4e00\u4e2a\u9636\u4e3a$M$\u6307\u6570\u4e3a$k$\u7684\u5206\u679d\u70b9\u7684\u66f2\u9762\u7ed5\u4e86$M+1$\u6b21\u4ee5\u53ca\u4e0a\u4e0b\u7a7f\u8fc7$M+k+1$\u6b21. \u5f53\u5206\u679d\u70b9\u5728\u8fb9\u754c\u65f6, \u6211\u4eec\u53ea\u770b\u5230\u8be5\u66f2\u9762\u7684\u4e00\u534a. \u56e0\u6b64, \u5f53\u66f2\u9762\u4e0e\u8fb9\u754c\u5355\u8c03\u5730\u63a5\u89e6\u65f6, \u8fb9\u754c\u5206\u679d\u70b9\u7684\u9636\u5fc5\u4e3a\u5076\u6570; \u5f53\u66f2\u9762\u4e0e\u8fb9\u754c\u63a5\u89e6\u662f\u4e24\u6b21\u7ed5\u56de\u65f6, \u8fb9\u754c\u5206\u679d\u70b9\u7684\u9636\u5fc5\u4e3a\u5947\u6570. \u56e0\u6b64, \u9636\u4e3a2\u65f6\u662f\u6700\u7b80\u5355\u7684\u60c5\u5f62. \u6307\u6570\u4e3a0\u5c06\u7ed9\u51fa\u5e73\u9762\u7684\u4e00\u90e8\u5206, \u800c\u6307\u6570\u4e3a1\u662f\u6700\u7b80\u5355\u7684\u4e09\u7ef4\u60c5\u5f62. \u800c\u9636\u4e3a2, \u6307\u6570\u4e3a1\u4f5c\u4e3a\u5185\u90e8\u5206\u679d\u70b9, \u5c06\u7ed53\u6b21, \u800c\u4e14\u4e0a\u4e0b\u7a7f\u8fc74\u6b21; \u800c\u4f5c\u4e3a\u8fb9\u754c\u5206\u679d\u70b9\u7684\u8bdd, \u5b83\u5c06\u7ed51.5\u6b21, \u4e0a\u4e0b\u7a7f\u8fc72\u6b21. \u4e00\u4e2a4\u9636\u6307\u6570\u4e3a2\u7684\u8fb9\u754c\u5206\u679d\u70b9, \u5c06\u7ed52.5\u6b21, \u4e0a\u4e0b\u7a7f\u8fc73.5\u6b21. \u8fd9\u4e9b\u6b63\u662f\u4e0a\u9762\u5404\u56fe\u6240\u793a\u7684\u60c5\u51b5. <\/p>\n\n\n\n<p>\u4e0a\u9762\u5404\u4e2a\u793a\u4f8b\u7684\u56fe\u7247\u4e2d\u7684\u66f2\u9762, \u53ef\u4ee5\u5229\u7528\u590d\u53d8\u91cf$z$\u7528\u5982\u4e0b\u7b80\u5355\u516c\u5f0f\u523b\u753b. \u5373, \u5982\u679c\u66f2\u9762\u7531\u4e09\u4e2a\u53c2\u6570\u7ed9\u51fa$\\{X(z),Y(z),Z(z)\\}$, \u90a3\u4e48\u5b83\u4eec\u6ee1\u8db3<\/p>\n\n\n\n<p>\\begin{align*} <br>X+iY&amp;=z^{M+1}\/(M+1);\\\\<br> Z&amp;=\\mathrm{Re}[z^{M+k+1}\/(M+k+1)] <br>\\end{align*}<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u516c\u5f0f\u4e2d, $\\mathrm{Re}$\u8868\u793a\u4e00\u4e2a\u590d\u6570\u7684\u5b9e\u90e8, \u800c\u7b2c\u4e00\u4e2a\u516c\u5f0f\u8868\u660e\u53f3\u7aef\u590d\u6570\u7684\u5b9e\u90e8\u548c\u865a\u90e8\u5206\u522b\u4e3a$X,Y$. \u6ce8\u610f\u5230\u5982\u679c$z$\u7684\u6781\u5750\u6807\u4e3a$(r,\\theta)$, \u90a3\u4e48<br>\\begin{align*}<br>X&amp;=\\frac{1}{M+1}r^{M+1}\\cos((M+1)\\theta)\\\\<br>Y&amp;=\\frac{1}{M+1}r^{M+1}\\sin((M_1)\\theta),<br>\\end{align*}<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u516c\u5f0f\u53ef\u4ee5\u7cbe\u786e\u8bf4\u660e\u524d\u9762\u8bf4\u7684\u7ed5\u51e0\u5708. \u4e8b\u5b9e\u4e0a, \u4e0a\u9762\u7684\u56fe\u7247\u6211\u4eec\u663e\u793a\u7684\u662f\u7684\u66f2\u9762\u5b9a\u4e49\u5728\u5706\u76d8$|z|\\leq 0.6$\u7684\u8fd9\u4e00\u90e8\u5206.<\/p>\n\n\n\n<p>\u5bf9\u521d\u5b66\u8005\u4e00\u4e2a\u6709\u610f\u601d\u7684\u95ee\u9898\u662f: \u5bf9\u66f2\u9762\u7684\u4e00\u4e2a\u9636\u4e3a$M$\u6307\u6570\u4e3a$k$\u7684\u5206\u679d\u70b9, \u66f2\u9762\u7684\u81ea\u4ea4\u7ebf\u6709\u591a\u5c11\u6761?<\/p>\n\n\n\n<p>\u4efb\u4f55\u4e00\u4e2a\u5206\u679d\u70b9\u5fc5\u88ab\u7c7b\u4f3c\u5730\u4e0a\u8ff0\u516c\u5f0f\u523b\u753b. \u4f46\u5bf9$z$\u7684\u9ad8\u9636\u5e42\u6b21, \u53ef\u80fd\u6709\u4e00\u4e9b\u65b0\u7684\u672f\u8bed. \u5982\u679c$M+1$\u6b63\u597d\u6574\u9664$M+k+1$, \u90a3\u4e48\u4f5c\u4e3a\u4e00\u6b21\u8fd1\u4f3c, \u66f2\u9762\u5c06\u662f\u5468\u671f\u91cd\u5408\u7684(\u5982\u679c\u4f60\u5c1d\u8bd5\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898, \u90a3\u4e48\u4f60\u4f1a\u6ce8\u610f\u5230\u8fd9\u4e2a\u4e8b\u5b9e). \u5982\u679c\u66f2\u9762\u53ef\u4ee5\u5206\u5272\u6210\u5404\u4e2a\u66f2\u9762\u7247, \u90a3\u4e48\u8be5\u66f2\u9762\u5fc5\u987b\u662f\u7531\u4e0a\u8ff0\u516c\u5f0f\u4e2d\u9ad8\u6b21\u5e42\u7ed9\u51fa\u7684.  \u4f60\u53ef\u4ee5\u770b\u5230\u4e0b\u56fe\u4e2d, \u5b83\u8868\u793a\u8fb9\u754c\u5206\u679d\u70b9\u7684\u9636\u4e3a2\u6307\u6570\u4e3a3\u7684\u60c5\u5f62. \u56e0\u6b64$M+1=3$, $M+k+1=6$. \u6ce8\u610f\u521d\u770b\u66f2\u9762\u50cf\u662f\u626d\u66f2\u7684\u5706\u76d8; \u4f46\u662f\u4ed4\u7ec6\u89c2\u5bdf, \u901a\u8fc7\u66f2\u9762\u7684\u5750\u6807\u6807\u8bb0, \u53d1\u73b0\u66f2\u9762\u4e8b\u5b9e\u4e0a\u7ed5\u4e861.5\u6b21, \u5b83\u7684\u540e\u534a\u6b21\u548c\u7b2c\u4e00\u4e2a\u534a\u6b21\u5b8c\u5168\u91cd\u5408.<\/p>\n\n\n\n<p><figure class=\"wp-block-image\">\n<center><a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order2Index3.gif\"><img decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order2Index3.gif\" alt=\"Order2Index3.gif\"><\/a>\n<\/center>\n<figcaption>\u56fe5. \u66f2\u9762\u7684\u4f2a\u5206\u679d\u70b9: \u96362\u6307\u65701<\/figcaption><\/figure><\/p>\n\n\n\n<p>\u5982\u679c\u4e0a\u8ff0\u91cd\u5408\u662f\u7cbe\u786e\u7684, \u6b63\u5982\u4e0a\u56fe\u6240\u793a, \u5206\u679d\u70b9\u79f0\u4e3a\u662f\u4f2a\u5206\u679d\u70b9. \u4f46\u4e00\u5b9a\u8981\u6ce8\u610f\u66f2\u9762\u770b\u8d77\u6765\u51e0\u4e4e\u5982\u6b64, \u66f2\u9762\u7684\u88ab\u5f88\u7ec6\u7684\u91cf\u5206\u5272\u6210\u5404\u4e2a\u7247(\u5728\u8ba1\u7b97\u673a\u4f5c\u56fe\u4e2d\u751a\u81f3\u4e0d\u53ef\u89c1), \u8fd9\u662f\u56e0\u4e3a\u5b83\u4eec\u662f\u7531$z$\u7684\u9ad8\u6b21\u5e42\u7684\u9879\u51b3\u5b9a\u7684. \u4e00\u4e2a\u5206\u679d\u70b9\u4e0d\u662f\u4f2a\u5206\u679d\u70b9\u5c31\u79f0\u4e3a\u771f\u5206\u679d\u70b9.<\/p>\n\n\n<p>\u4e0b\u56fe\u662f\u4e00\u4e2a\u66f2\u9762\u7684\u66f4\u52a0\u590d\u6742\u7684\u5206\u679d\u70b9<\/p>\n<figure class=\"wp-block-image\">\n<center><a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order4Index2.gif\"><img decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2021\/05\/Order4Index2.gif\" alt=\"Order4Index2.gif\"><\/a><br>\n<\/center><figcaption>\u56fe6. \u66f2\u9762\u7684\u4f2a\u5206\u679d\u70b9: \u96362\u6307\u65701<\/figcaption><\/figure>","protected":false},"excerpt":{"rendered":"<p>\u6781\u5c0f\u66f2\u9762\u7684\u5206\u679d\u70b9\u5b9a\u4e49\u4e3a\u66f2\u9762\u5728\u8fd9\u6837\u7684\u70b9\u5904\u4e0d\u80fd\u7528\u4e00\u5bf9\u4e00\u7684\u65b9\u5f0f\u53c2\u6570\u5316. \u4ece\u800c\u5728\u5206\u679d\u70b9, \u66f2\u9762\u5e76\u4e0d\u662f\u901a\u5e38\u5b9a\u4e49\u4e0b\u7684\u51e0\u4f55\u66f2&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=756\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6781\u5c0f\u66f2\u9762\u7684\u5206\u679d\u70b9(branched point)<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-756","post","type-post","status-publish","format-standard","hentry","category-math","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/756","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=756"}],"version-history":[{"count":12,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/756\/revisions"}],"predecessor-version":[{"id":854,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/756\/revisions\/854"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=756"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=756"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}