What is the difference between:
a) a complex differentiable at a point $z$ or
b) a holomorphic at a point $z$,
for a function $f(z)$, on the complex plane?
It was my understanding that holomorphicity is just stronger than differentiability i.e. everywhere in the neighbourhood of that point is complex differentiable.
How do you go about showing where a complex function is either complex differentiable or holomorphic?
Any help would be greatly appreciated!
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.
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.
