《第五章解析函数罗的朗展式.ppt》由会员分享，可在线阅读，更多相关《第五章解析函数罗的朗展式.ppt（30页珍藏版）》请在三一文库上搜索。
1、西南科技大学大学本科理科数学类专业课程 第五章 解析函数的罗朗展式 v一、双边幂级数 v二、解析函数的罗朗展式 v三、解析函数的孤立奇点及其分类 v四、解析函数在无穷远点的性质 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2
2、011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数12007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程
3、 第一节、解析函数的洛朗展式 v1、定义 形如 被称为关于的双边幂级数。 一、双边幂级数 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluati
4、on only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数22007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 则有 在H内绝对收敛且内闭一致收敛于 在H内逐项求导p次，p=1,2, 2、性质 定理
5、5.1 设双边幂级数的收敛圆环为 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose
6、.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数32007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 r R p 展开式是唯一的 可展开成双边幂级数（关于即 其中 若则 二、定理5.2(Laurent定理) (*) Evaluation onl
7、y.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Pro
8、file 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数42007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 注： （1）若 在 内能展开成Laurent级数; * （3）若则展开式必为如下形式： 表示成 （4）展开法-根据展开式唯一，只须将 (*)即为Laurent展开式。 （2）Taylor级数与Lauren
9、t级数的关系;Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides fo
10、r .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数52007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 （1） （2） 解： 例1.求函数在下列区域内的罗朗展式 （1） 在内解析 内可展开成罗朗级数，即在 Evaluation only.Evaluation onl
11、y. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Cre
12、ated with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数62007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 在 在 （1） 内解析 内可展开成罗朗级数，即 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.
13、0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Cop
14、yright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数72007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 同理可得 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-20
15、11 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数82007.
16、09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 其级数形式为 令 则 （2） 在内解析 在内可展成罗朗级， Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Eval
17、uation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数92007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 即(1) 又因 Evaluation only
18、.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Prof
19、ile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数102007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 (2) 由（1）、（2）可知 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile
20、 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0
21、.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数112007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 在 处的去心邻展成罗朗级数。 解： 以和为奇点 的的去心邻域为 令则有 且 例2. 将 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NE
22、T 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copy
23、right 2004-2011 Aspose Pty Ltd. 课程名称：复变函数122007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 对于有 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-201
24、1 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数132007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程
25、Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET
26、3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数142007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 设a为的孤立奇点，即 使在去心 邻域内可以展开成Laurent级数。 此时，为在a的正则部分。 为在点a处的主要部分。 第二节、解析函数的孤立奇点及其分类 若在点a处的主要部
27、分为零， 的可去奇点； 则称为 若在点a处的主要部分为 的m级极点； 则称 为 若在点a处的主要部分有无穷多项， 的本性奇点； 则称 为 1、定义5.3 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose P
28、ty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数152007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 定理5.3为设的孤立
29、奇点，则 的可去点;为 在的主要部分为0; 在点a的去心邻域内有界。 2、各类孤立奇点的特征 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evalua
30、tion only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数162007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 解： 的可去奇点为 解： 例1.判别下列函数在指定奇点处的类型 为的可奇点。 E
31、valuation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3
32、.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数172007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 设以为孤立奇点，则下列命题等价 以 为m级极点; 在处的主要部分： 定理5.4 在点a处解析， 为m级零点;以 Evaluation only.Evaluation only.
33、 Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Creat
34、ed with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数182007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 解： 仅以为奇点 例2. 判断下列函数在指定点处的性质 在处解析，且 记 的孤立奇点。为 为的一级极点。 Evaluation only.Evaluation only. Created with Aspose.Slides for .N
35、ET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5
36、 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数192007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 解：以 z = 0, 1 为奇点 为的孤立奇点。 例2. 判断下列函数在指定点处的性质 我们有对于 为的二级极点。 在处解析， Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5
37、.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0
38、. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数202007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 在处解析， 同理我们有对于 为的一级极点。 定理5.5 设 的某邻域内不恒等于零 ，则 在 以 为 m 级极点 以 为 m 级零点 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with A
39、spose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011
40、 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数212007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 解： 记则 所以存在 为一级零点以 处解析，且在 处的函数值不等于 零的函数使得 为的二级零点。 在处解析，且 为故为二级极点。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET
41、3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyri
42、ght 2004-2011 Aspose Pty Ltd. 课程名称：复变函数222007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 不存在且不为 定理5.7为的本性奇点，且在点a的充分小去 的本性奇点 心邻域内不为零，则 也必为 定理5.6为的本性奇点 本性奇点的特征： Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profil
43、e 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Asp
44、ose Pty Ltd. 课程名称：复变函数232007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 不存在且不为 为故的本性奇点。 不存在且不为 的本性奇点 所以由定理5.7可知 为 即 也为的本性奇点。 均为 与的本性奇点. 证明:为 的本性奇点. 例3. 证明： Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5
45、.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose
46、 Pty Ltd. 课程名称：复变函数242007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 分析： 在 关键是讨论是存在使 内解析。 解： 下列函数是否以例4.为孤立点？ 在有限复平面上仅以为奇点 在内解析 为的孤立奇点。 为的孤立奇点的定义： 内解析，则称为的一个孤立奇点。 定义5.4 若函数 在无穷远点（去心）邻域 第三节、解析函数在无穷远点的性质 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created wit
47、h Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2
48、011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数252007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 为奇点在有限复平面内以解： 而 时, 的非孤立奇点故为 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyri
49、ght 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 课程名称：复变函数262007.09.01 主讲教师：卢 谦 西南科技大学大学本科理科数学类专业课程 (2) 无穷远点的分类与判定方法 在为孤立奇点，则设以 内解析，令则在函数 为 的孤立奇点。 在内解析，即 的可去奇 点，m级极点和本性奇点， 则相应地 的可去奇点，m级极点和本 性奇点。 为 为 Evaluation only.Evaluation only.