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所在平台: Udemy |
课程主页: https://www.udemy.com/course/conformal-transformations-complex-analysis/
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**课程名称：** 保形映射 1 (复分析) **课程概述：** 本课程深入探讨保形映射（又称保形变换、角度保持变换、双全纯映射），这是一种能够保持局部角度的变换。课程将带领学员在 z 平面和 w 平面之间进行点映射，内容涵盖： * **变换的概念与雅可比行列式：** 详细讲解变换的原理以及雅可比行列式的意义。 * **区域映射：** 学习如何确定 w 平面中与 z 平面中给定区域相对应的区域。 * **保形映射的充要条件：** 明确 w = f(z) 代表保形映射的必要及充分条件。 * **表观放大率与圆的反演点：** 探讨圆的相关概念，如度量放大率和反演点。 * **基本变换：** 涵盖平移变换、旋转变换、放大变换、旋转与放大联合变换、反演变换、线性变换等基础的映射类型。 * **双线性变换（分数线性变换）：** 深入研究双线性变换，包括其行列式、标准化形式、莫比乌斯变换及其关键点。 * **变换的复合：** 讲解变换的叠加（乘积）运算。 * **双线性变换与交比：** 重点阐述双线性变换如何保持交比不变。 * **确定双线性变换：** 教授如何确定将 z 平面上的点映射到 w 平面的双线性变换。 * **斯泰纳圆与圆簇：** 探讨斯泰纳圆以及圆簇之间的关系。 * **双线性变换的标准形式与不动点：** 讲解双线性变换的标准形式以及其不动点的概念。 * **变换的映射性质：** 重点证明双线性变换可以将圆或直线映射为圆或直线，并将反演点映射为反演点。 * **多种变换类型：** 涵盖椭圆变换、双曲变换、抛物线变换及趋同变换（loxodromic transformation），并包含所有预期的已解示例和重要定理。
A Conformal Mapping, also called a Conformal Map, Conformal Transformation, Angle-preserving transformation, or Biholomorphic map, is a transformation that preserves local angles. The Course 'Conformal Transformations' describes about the mapping of points in the z plane to w plane including the other contents_Detailed concept of Transformations and Jacobian of Transformation.To determine the region in the w plane corresponding to the region given in z plane.Necessary and Sufficient Condition for w = f(z) to represent Conformal Mapping.Superficial magnification and Inverse points with respect to a Circle.Some Elementary Transformation as Translation Transformation, Rotation Transformation, Magnification Transformation, Rotation and magnification Transformation, Inversion Transformation, Linear transformation.Bilinear or Linear Fractional Transformation.Determinant of Transformation and its Normalized Form.Mobius Transformation and Critical Points.Resultant or Product of Transformation.Preservance of Cross ratio under bilinear TransformationTo Determine the Bilinear Transformation which maps the points in z plane to the points in w plane.Steiner Circles and Family of circles.Normal Form of Bilinear transformation and Fixed Points of Bilinear Transformation.Every Bilinear Transformation transforms circles or straight lines into circles or straight lines and inverse points into inverse points.Elliptical Transformation, Hyperbolic Transformation, Parabolic Transformation & Loxodromic Transformation including all expected solved examples and Important Theorems.
