4
votes
2answers
82 views

A question on notation: $u(x,y)$ vs. $u(x+iy)$.

In Tao's complex analysis notes, he uses the convention that $$f(z)=u(z)+iv(z)= u(x+iy)+iv(x+iy)$$ implying that the functions are $$u,v:\mathbb{C} \to \mathbb{R}.$$ But in other places I see the ...
1
vote
4answers
113 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...
0
votes
1answer
171 views

Square root principle value convention

Why is the principal square root of a complex number defined as $\sqrt z = \sqrt r e^{-i \varphi / 2}$ for $\varphi \in (-\pi, \pi]$ ? Wouldn't it be more natural to let $\varphi \in [0, 2\pi)$ as it ...