What is the difference between complex differentiable and holomorphic functions at a point?

Solution 1:

In principle, you check the function is continuous and has partial derivatives in the region, and that the Cauchy-Riemann equations hold.

In practice, depending on how the function is defined, you usually rely on results that tell you that various operations on holomorphic functions produce holomorphic functions.

Solution 2:

As a matter of fact, if a complex function is differentiable on $\mathbb C$ (i.e., $f'(z)$ exists and it is continuous for every $z\in\mathbb{C}$), then it is also analytic on $\mathbb C$, that is, it is $C^\infty$ and admits a power series representation. This follows from the Cauchy integral formula, the cornerstone of complex analysis.

Note: It actually turns out that the continuity of $f'(z)$ is not necessary, but it actually follows from its existence. This is also called Goursat's lemma, which states that if $f(z)$ is differentiable in the complex sense, then $f'(z)$ is continuous.

Holomorphic functions are rare, but as such enjoy some tremendous properties that we can only dream about for functions of real variables.