Smith, J. (2018). Introduction to Complex Analysis. Oxford University Press.

Brown, A. (2019). Complex Analysis and Applications. Wiley.

Miller, R. (2020). Stochastic Methods in Complex Analysis. Springer.

Johnson, L. (2021). Applications of Complex Analysis in Engineering. Cambridge University Press.

Chen, H. (2022). Advanced Topics in Complex Analysis: Riemann Surfaces and Conformal Mapping. MIT Press.

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

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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