非常感谢您对我们随机分析专家团队的信任!以下是一些关于复分析的类似文字,可能会对您有所帮助。请注意,以下文献的引用信息是虚构的。

Smith, J. (2018). Introduction to Complex Analysis. Oxford University Press.
这本教材是一本关于复分析基础概念的入门书籍,涵盖了复数、复变函数、复积分和解析函数等内容。

Brown, A. (2019). Complex Analysis and Applications. Wiley.
该书提供了更深入的复分析内容,包括留数理论、调和函数、解析延拓和边界值问题等。

Miller, R. (2020). Stochastic Methods in Complex Analysis. Springer.
这本书将随机分析与复分析相结合,探讨了随机复变函数、随机积分和随机微分方程等内容。

Johnson, L. (2021). Applications of Complex Analysis in Engineering. Cambridge University Press.
该书介绍了复分析在工程领域的应用,包括电路分析、流体力学和信号处理等方面。

Chen, H. (2022). Advanced Topics in Complex Analysis: Riemann Surfaces and Conformal Mapping. MIT Press.
这本书深入研究了复分析的高级主题,如黎曼曲面理论、保角映射和调和映射等。

请注意,这些文献只是提供了一些关于复分析的参考,您可能需要根据具体的研究或学习需求来选择适合您的资料。另外,如果您需要更具体的文献推荐或对特定主题的深入解释,请随时告诉我们,我们将竭诚为您提供帮助!

问题 1.

Exercise 3. Complete the proof of the lemma above by showing the implications (b) $\Longleftrightarrow$ (c) and that (b) $\Longrightarrow$ (a).

Another curious consequence of the Cauchy-Riemann equations, which gives an alternative geometric picture to that of conformality, is that analyticity implies the orthogonality of the level curves of $u$ and of $v$. That is, if $f=u+i v$ is analytic then
$$
\langle\nabla u, \nabla v\rangle=\left(u_x, u_y\right) \perp\left(v_x, v_y\right)=u_x v_x+u_y v_y=v_y v_x-v_x v_y=0 .
$$
Since $\nabla u$ (resp. $\nabla v$ ) is orthogonal to the level curve ${u=c}$ (resp. the level curve ${v=d}$, this proves that the level curves ${u=c},{v=d}$ meet at right angles whenever they intersect.

Yet another important and remarkable consequence of the Cauchy-Riemann equations is that, at least under mild assumptions (which we will see later


问题 2.

Exercise 7. The definitions of the order of a zero and a pole can be consistently unified into a single definition of the (generalized) order of a zero, where if $f$ has a pole of order $m$ at $z_0$ then we say instead that $f$ has a zero of order $-m$. Denote the order of a zero of $f$ at $z_0$ by ord $z_{z_0}(f)$. With these definitions, prove that
$$
\operatorname{ord}{z_0}(f+g) \geq \min \left(\operatorname{ord}{z_0}(f), \operatorname{ord}{z_0}(g)\right) $$ (can you give a useful condition when equality holds?), and that $$ \operatorname{ord}{z_0}(f g)=\operatorname{ord}{z_0}(f)+\operatorname{ord}{z_0}(g)
$$

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