论文标题
复杂分析功能和矢量场的先验奇异性的几何形状
Geometry of transcendental singularities of complex analytic functions and vector fields
论文作者
论文摘要
在Riemann表面$ m $上，存在可能的多值函数$ψ_x$之间的规范对应关系，该功能的差异是单个值（$ $ $ a添加性自动形态上的复杂分析功能）和矢量字段$ x $。从向量场的角度来看，我们认为的奇异性是零，杆，孤立的基本奇异性和上述积累点。逆函数的奇异性$ψ_x^{ - 1} $从Meromorphic函数扩展到加上自动型奇异复杂分析函数。主要的贡献是$ψ_x^{ - 1} $的奇异性是代数，对数或来自零零残基的零零，$ x $的奇异性是一个完整的表征。出现了$ψ_x$的分析属性，$ψ_x^{ - 1} $的奇异性与$ x $的奇异性之间的关系。研究了$ψ_x^{ - 1} $奇异性附近的矢量场的几何丰富度的家庭和零星例子。作为应用程序；描述了矢量字段$ x $的复杂轨迹解决方案的最大统一区域，对矢量字段$ x $不完整的真实轨迹的几何表征以及与riemann $ξ$函数相关的矢量场奇点的描述。
On Riemann surfaces $M$, there exists a canonical correspondence between a possibly multivalued function $Ψ_X$ whose differential is single valued ($i.e.$ an additively automorphic singular complex analytic function) and a vector field $X$. From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities and accumulation points of the above. The theory of singularities of the inverse function $Ψ_X^{-1}$ is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of $Ψ_X^{-1}$ is either algebraic, logarithmic or arises from a zero with nonzero residue of $X$. Relationships between analytical properties of $Ψ_X$, singularities of $Ψ_X^{-1}$ and singularities of $X$ are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of $Ψ_X^{-1}$ are studied. As applications we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field $X$, a geometric characterization of the incomplete real trajectories of a vector field $X$, and a description of the singularities of the vector field associated to the Riemann $ξ$ function.
