LEC CANA 1

Stein & Shakarchi (2003) chapter 1.

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[!Definition]
$f:\mathbb{C}\to \mathbb{C}$ is said to be holomorphic at a point $z$ if

$$\underset{h\to 0}{\lim }\frac{f(z+h)-f(z)}{h}$$

exists.

$f$ is holomorphic at $p$ $⟺$ $f$ has a power series expansion at $p$, $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$. $f$ can be viewed at $u+iv$, where $u$ and $v$ are functions ${\mathbb{R}}^2\to \mathbb{R}$.

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References

Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press.