Analytic and holomorphic functions
Analytic, or holomorphic, functions are functions f(z) defined on an open set \mathcal{D} \subseteq \mathbb{C} that are single-valued and admit a finite complex derivative at every point in \mathcal{D}. The complex derivative is given by:
f^\prime(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}
where the limit must exist uniquely, independent of the way \Delta z \to 0 in the complex plane.
A property of analytic functions is that they are infinitely differentiable and possess a convergent Taylor series expansion around any point in their domain:
f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n
with f^{(n)}(z_0) denoting the n-th complex derivative at z_0.
Examples of Analytic Functions
Classical examples include polynomials, such as f(z) = z^n, and entire functions like the exponential f(z) = e^z and the trigonometric functions f(z) = \sin(z) and f(z) = \cos(z).
Rational functions R(z) = P(z)/Q(z), where P and Q are polynomials and Q(z) \neq 0, are analytic wherever the denominator is nonzero.
The principal branch of the complex logarithm f(z) = \mathrm{Log}(z) is analytic in its natural domain, for example \mathbb{C} \setminus (-\infty, 0].
For example, f(z) = z^2,
f^\prime(z) = \lim_{\Delta z \to 0} \frac{(z + \Delta z)^2 - z^2}{\Delta z} = \lim_{\Delta z \to 0} (2z + \Delta z) = 2z
regardless of the path, showing the derivative exists everywhere and is unique.
Examples of Functions That Are Not Analytic
Some common functions are not analytic at any point.
The complex conjugate f(z) = \bar{z}, as well as real and imaginary part functions f(z) = \Re(z) and f(z) = \Im(z), and modulus f(z) = |z|, do not have a well-defined complex derivative.
In these cases, the limit above is path-dependent or undefined. Additionally, multivalued functions such as f(z) = z^a for non-integer a are not, in general, analytic unless restricted to a specific branch.
For example, f(z) = \bar{z},
f^\prime(z) = \lim_{\Delta z \to 0} \frac{\overline{\Delta z}}{\Delta z}
This limit depends on the direction: along the real axis the limit is 1, and along the imaginary axis it is -1. Thus, the complex derivative does not exist at any point.
For more insights into this topic, you can find the details here.