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1、OutlineOutline第 四 讲无 穷 级 数1OutlineOutline 讲授要点讲授要点 复数级数复数级数 复数级数 收敛与发散 绝对收敛 函数级数函数级数 函数级数的收敛性 函数级数的一致收敛性 幂级数幂级数 Abel 原理 收敛圆与收敛半径2OutlineOutline 讲授要点讲授要点 Taylor 展开展开 展开定理 讨论 基本展开式 Laurent 展开展开 展开定理及讨论 例子 3Complex seriesComplex seriesSeries of Complex FunctionSeries of Complex FunctionPower SeriesPowe
2、r Series4Complex Series Complex Series Convergence and divergenceConvergence and divergence5Complex Series Complex Series Convergence and divergenceConvergence and divergence6Complex Series Complex Series Convergence and divergenceConvergence and divergence7Complex Series Complex Series Convergence
3、and divergenceConvergence and divergence8Complex Series Complex Series Absolutely ConvergenceAbsolutely Convergence9Complex Series Complex Series Absolutely ConvergenceAbsolutely Convergence10Series of Complex Functions Series of Complex Functions ConvergenceConvergence11Series of Complex Functions
4、Series of Complex Functions ConvergenceConvergence 函数级数的收敛性函数级数的收敛性12Series of Complex Functions Series of Complex Functions ConvergenceConvergence函数级数的一致收敛性函数级数的一致收敛性13OutlineOutline 一致收敛性判别法一致收敛性判别法14Power SeriesPower SeriesAbel TheoremsAbel TheoremsCircle of Convergence and It RadiusCircle of Con
5、vergence and It Radius15Power SeriesPower SeriesAbel TheoremsAbel TheoremsCircle of Convergence and It RadiusCircle of Convergence and It Radius16Power SeriesPower SeriesAbel TheoremsAbel TheoremsCircle of Convergence and It RadiusCircle of Convergence and It Radius17Power SeriesPower SeriesAbel The
6、oremsAbel TheoremsCircle of Convergence and it RadiusCircle of Convergence and it Radius收敛半径的求法收敛半径的求法18Power SeriesPower SeriesAbel TheoremsAbel TheoremsCircle of Convergence and it RadiusCircle of Convergence and it Radius收敛半径的求法收敛半径的求法19Power SeriesPower SeriesAbel TheoremsAbel TheoremsCircle of
7、Convergence and it RadiusCircle of Convergence and it Radius收敛半径公式讨论收敛半径公式讨论20Expansion in Taylor seriesExpansion in Taylor series21Expansion in Taylor seriesExpansion in Taylor seriesTheorem (Taylor)Theorem (Taylor)22Expansion in Taylor seriesExpansion in Taylor seriesTheorem (Taylor)Theorem (Taylo
8、r)23Expansion in Taylor seriesExpansion in Taylor seriesTheorem (Taylor)Theorem (Taylor)24Expansion in Taylor seriesExpansion in Taylor seriesTheorem (Taylor)Theorem (Taylor)25Expansion in Taylor seriesExpansion in Taylor seriesTheorem (Taylor)Theorem (Taylor)26Expansion in Taylor seriesExpansion in
9、 Taylor seriesDiscussions and RemarksDiscussions and Remarks讨论讨论27Expansion in Taylor seriesExpansion in Taylor series Discussions and RemarksDiscussions and Remarks讨论讨论28Expansion in Taylor seriesExpansion in Taylor series Discussions and RemarksDiscussions and Remarks讨论讨论29Expansion in Taylor seri
10、esExpansion in Taylor seriesDiscussions and RemarksDiscussions and Remarks举例举例30Expansion in Taylor seriesExpansion in Taylor series Discussions and RemarksDiscussions and Remarks讨论讨论31Expansion in Taylor seriesExpansion in Taylor series Discussions and RemarksDiscussions and Remarks32Expansion in T
11、aylor seriesExpansion in Taylor seriesIllustrative ExamplesIllustrative Examples基本函数展开式基本函数展开式33Taylor Expansion: ExamplesTaylor Expansion: Examples34Taylor Expansion: ExamplesTaylor Expansion: Examples35Taylor Expansion: ExamplesTaylor Expansion: ExamplesProduct of Two SeriesProduct of Two Series 级
12、数乘法级数乘法36Taylor Expansion: ExamplesTaylor Expansion: ExamplesProduct of Two SeriesProduct of Two Series 待定系数法待定系数法37Taylor Expansion: ExamplesTaylor Expansion: ExamplesProduct of Two SeriesProduct of Two Series 待定系数法待定系数法38Taylor Expansion: ExamplesTaylor Expansion: ExamplesProduct of Two SeriesProd
13、uct of Two Series 待定系数法待定系数法39Taylor Expansion: ExamplesTaylor Expansion: ExamplesProduct of Two SeriesProduct of Two Series 待定系数法待定系数法40Expansion in Laurent seriesExpansion in Laurent series41Expansion in Laurent seriesExpansion in Laurent seriesTheoremTheorem（LaurentLaurent）42Expansion in Laurent
14、seriesExpansion in Laurent seriesTheoremTheorem（LaurentLaurent）43Expansion in Laurent seriesExpansion in Laurent seriesTheoremTheorem（LaurentLaurent）44Expansion in Laurent seriesExpansion in Laurent seriesTheoremTheorem（LaurentLaurent）45Expansion in Laurent seriesExpansion in Laurent seriesTheorem (
15、Laurent)Theorem (Laurent)46Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)讨论讨论47Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)讨论讨论48Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laure
16、nt)讨论讨论49Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)讨论讨论50Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)讨论讨论51Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)无穷远点的无穷远点的Laure
17、nt展开展开52Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)无穷远点的无穷远点的Laurent展开展开53Expansion in Laurent seriesExpansion in Laurent seriesTheorem (Laurent)Theorem (Laurent)54Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Exa
18、mples55Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples56Expansion in Laurent seriesExpansion in Laurent series57Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples58Expansion in Laurent seriesExpansion in L
19、aurent seriesIllustrative ExamplesIllustrative Examples讨论讨论59Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples讨论讨论60Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples61Expansion in Laurent seriesExpansion in
20、 Laurent seriesIllustrative ExamplesIllustrative Examples62Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples63Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples64Expansion in Laurent seriesExpansion in Laure
21、nt seriesIllustrative ExamplesIllustrative Examples65Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples66Expansion in Laurent seriesExpansion in Laurent seriesIllustrative ExamplesIllustrative Examples67Expansion in Laurent seriesExpansion in Laurent ser
22、iesIllustrative ExamplesIllustrative Examples评述评述68Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities69Isolated Singularities of Isolated Singularities of Uniform FunctionUniform Fun
23、ctionIsolated SingularityIsolated Singularity70Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities可去奇点可去奇点71Isolated Singularities of Isolated Singularities of Uniform FunctionUniform
24、 FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities函数在可去奇点处的行为函数在可去奇点处的行为72Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities函数在可去奇点处的行为函数在可
25、去奇点处的行为73Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities函数在可去奇点处的行为函数在可去奇点处的行为74Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Feature
26、s of Isolated Typical Features of Isolated SingularitiesSingularities极点极点75Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities函数在极点处的行为函数在极点处的行为76Isolated Singularities of Isolated Si
27、ngularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities函数在极点处的行为函数在极点处的行为77Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities本性奇点本性奇点78Isolated Singularities of Isolated Singularities of Uniform FunctionUniform FunctionTypical Features of Isolated Typical Features of Isolated SingularitiesSingularities总结总结79
