数学物理方法Chapt 2 _1.pptx

Contents of Chapt. 2:Singularities奇点Calculus of Residues学完第二章再一起复习这两章的内容Chapt 2: Calculus of Residuesshi.xq@sustc.edu.cnChapter 2: Calculus of Residues2.1 Singularities 奇点P497 of MMP多个概念Chapt 2: Calculus of Residuesshi.xq@sustc.edu.cnPolesIsolated singular point 孤立奇点If f (z) is not analytic at z = z0 but is analytic at all neighboring points, we define the point z0 as an isolated singular point of the function f (z). There will therefore be a Laurent expansion about an isolated singular point, and one of the following statements will be true:The most negative power of z-z0 in the Laurent expansion of f (z) about z-z0 will be some finite power, , where n is an integer, orThe Laurent expansion of f (z) about z-z0?will continue to negatively infinite powers of z-z0.Chapt 2: Calculus of Residuesshi.xq@sustc.edu.cn2.1 SingularitiesPolesPoles 极点Chapt 2: Calculus of Residuesshi.xq@sustc.edu.cn2.1 Singularities?In the first case , , the singularity is called a pole, and is more specifically identified as a pole of order n（n 阶极点）。A pole of order 1 is also called a simple pole. The second case (negatively infinite powers) is not referred to as a “pole of infinite order,” but is called an essential singularity （本性奇点）。PolesIdentify the order of a poleOne way to identify a pole of f (z) without having available its Laurent expansion is to examine The integer n for which a is finite and non-zero, gives the order of the pole at z = z0.This rule follows directly from the form of the Laurent expansion.If the above limit is equal to zero, then z = z0 is a pole of order less than n, or f(z) is analytic there; If the limit is infinite then the pole is of an order greater than n.for various positive integers n.写Chapt 2: Calculus of Residuesshi.xq@sustc.edu.cn2.1 SingularitiesPolesEssential singularities at z=0Essential singularities are often identified directly from their Laurent expansions. For example,which clearly has an essential singularity at z=0.Essential singularities hav