In complex analysis, an essential singularity of a function is a “severe” singularity near which the function exhibits odd behavior.

The category essential singularity is a “left-over” or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some[who?] include non-isolated singularities too; those do not have a residue.

Plot of the function exp(1/z), centered on the essential singularity at z = 0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).

Model illustrating essential singularity of a complex function 6w = exp(1/(6z))

1 Formal description
2 Alternative descriptions

1 Formal description

Consider an open subset {\displaystyle U}U of the complex plane {\displaystyle \mathbb {C} }\mathbb {C} . Let {\displaystyle a}a be an element of {\displaystyle U}U, and {\displaystyle f\colon U\setminus {a}\to \mathbb {C} }{\displaystyle f\colon U\setminus {a}\to \mathbb {C} } a holomorphic function. The point {\displaystyle a}a is called an essential singularity of the function {\displaystyle f}f if the singularity is neither a pole nor a removable singularity.

For example, the function {\displaystyle f(z)=e^{1/z}}{\displaystyle f(z)=e^{1/z}} has an essential singularity at {\displaystyle z=0}z=0.

2 Alternative descriptions

Let {\displaystyle ;a;}{\displaystyle ;a;} be a complex number, assume that {\displaystyle f(z)}f(z) is not defined at {\displaystyle ;a;}{\displaystyle ;a;} but is analytic in some region {\displaystyle U}U of the complex plane, and that every open neighbourhood of {\displaystyle a}a has non-empty intersection with {\displaystyle U}U.

If both {\displaystyle \lim {z\to a}f(z)}\lim{z \to a}f(z) and {\displaystyle \lim {z\to a}{\frac {1}{f(z)}}}\lim{z \to a}\frac{1}{f(z)} exist, then {\displaystyle a}a is a removable singularity of both {\displaystyle f}f and {\displaystyle {\frac {1}{f}}}{\displaystyle {\frac {1}{f}}}.
If {\displaystyle \lim {z\to a}f(z)}\lim{z \to a}f(z) exists but {\displaystyle \lim {z\to a}{\frac {1}{f(z)}}}\lim{z \to a}\frac{1}{f(z)} does not exist (in fact {\displaystyle \lim _{z\to a}|1/f(z)|=\infty }{\displaystyle \lim _{z\to a}|1/f(z)|=\infty }), then {\displaystyle a}a is a zero of {\displaystyle f}f and a pole of {\displaystyle {\frac {1}{f}}}{\displaystyle {\frac {1}{f}}}.
Similarly, if {\displaystyle \lim {z\to a}f(z)}\lim{z \to a}f(z) does not exist (in fact {\displaystyle \lim _{z\to a}|f(z)|=\infty }{\displaystyle \lim _{z\to a}|f(z)|=\infty }) but {\displaystyle \lim {z\to a}{\frac {1}{f(z)}}}\lim{z \to a}\frac{1}{f(z)} exists, then {\displaystyle a}a is a pole of {\displaystyle f}f and a zero of {\displaystyle {\frac {1}{f}}}{\displaystyle {\frac {1}{f}}}.
If neither {\displaystyle \lim {z\to a}f(z)}\lim{z \to a}f(z) nor {\displaystyle \lim {z\to a}{\frac {1}{f(z)}}}\lim{z \to a}\frac{1}{f(z)} exists, then {\displaystyle a}a is an essential singularity of both {\displaystyle f}f and {\displaystyle {\frac {1}{f}}}{\displaystyle {\frac {1}{f}}}.
Another way to characterize an essential singularity is that the Laurent series of {\displaystyle f}f at the point {\displaystyle a}a has infinitely many negative degree terms (i.e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point {\displaystyle a}a for which no derivative of {\displaystyle f(z)(z-a)^{{n}}f(z)(z-a)}n converges to a limit as {\displaystyle z}z tends to {\displaystyle a}a, then {\displaystyle a}a is an essential singularity of {\displaystyle f}f.[1]

On a Riemann sphere with a point at infinity, {\displaystyle \infty _{\mathbb {C} }}{\displaystyle \infty _{\mathbb {C} }}, the function {\displaystyle {f(z)}}{\displaystyle {f(z)}} has an essential singularity at that point if and only if the {\displaystyle {f(1/z)}}{\displaystyle {f(1/z)}} has an essential singularity at 0: i.e. neither {\displaystyle \lim _{z\to 0}{f(1/z)}}{\displaystyle \lim _{z\to 0}{f(1/z)}} nor {\displaystyle \lim _{z\to 0}{\frac {1}{f(1/z)}}}{\displaystyle \lim _{z\to 0}{\frac {1}{f(1/z)}}} exists.[2] The Riemann zeta function on the Riemann sphere has only one essential singularity, at {\displaystyle \infty _{\mathbb {C} }}{\displaystyle \infty _{\mathbb {C} }}.[3]

The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard’s great theorem. The latter says that in every neighborhood of an essential singularity {\displaystyle a}a, the function {\displaystyle f}f takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function {\displaystyle \exp(1/z)}{\displaystyle \exp(1/z)} never takes on the value 0.)

Essential singularity相关推荐

UA MATH524 复变函数13 奇点与留数
UA MATH524 复变函数13 奇点与留数零点的阶孤立奇点留数零点的阶假设fff满足 f(k)(z0)=0,k=0,1,⋯,m−1f(m)(z0)≠0f^{(k)}(z_0)=0,k=0 ...
signal(SIGPIPE, SIG_IGN)
关于SIGPIPE导致的程序退出当服务器close一个连接时,若client端接着发数据.根据TCP协议的规定,会收到一个RST响应,client再往这个服务器发送数据时,系统会发出一个SIGPIP ...
oracle求非偶非素数的和,Sub Maths__写给非数学专业的朋友们
[本文系寄托天下原创,转载请注明出处,拜托.] [综合版版主]GOGO ^^ 附4.GOGO的数学词汇 map image fixed point composite function one to ...
复变函数和积分变换(Complex Function II)
数学物理方法级数(Series) 复变函数项级数幂级数泰勒级数洛朗级数孤立奇点留数(Residue) 留数的概念与计算留数在定积分计算中的应用对数留数与辐角原理共形映射(Confor ...
关于长寿_你会长寿到永远吗
关于长寿 The fear of loneliness is a very common fear in life, and we are all experiencing that by being ...
Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually re ...
40奇点，天网，计算机的未来
计算机速成课第40集奇点,天网,计算机的未来超喜欢此系列, 整理成文字笔记方便重复阅读. 01:21 普适计算 Ubiquitous Computing 04:55 奇点 Singularity ...
[转载]常用数学专业名词的英语
目录前言数学分支的英文: 命题的英文: 数学中常见数词的英语: 点的英文: 线的英文: 面的英文: 角的英文: 距离的英文: 多边形的英文: 三角形的英文: 四边形的英文: 圆的英文: 多面体的英 ...
复变函数与积分变换中的英汉单词对照
复数与复变函数复数 complex number 实部 real number 虚部 imaginary unit 纯虚数 pure imaginary number 共轭复数 complex co ...

Essential singularity

Contents

1 Formal description

2 Alternative descriptions

Essential singularity相关推荐

最新文章

热门文章