What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that:

A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

Can you put this is simpler terms or illustrate it with an example? Also, it seems very similar to the definition of analytic, are they equivalent?

Solution 1:

In fact holomorphic functions on a domain $U$ are identical to analytic ones, but the definitions shouldn't look that similar. If you've seen a similar definition of "analytic," it was cheating: an analytic function $f$ on $U$ is just one with a power series, $f(z)=\sum a_i(z-a)^i$ for some $a\in U$. It's an important theorem that this is actually equivalent to complex differentiability on $U$, which is very far from being the case over $\mathbb{R}$.

The "simpler terms" you request may be covered by recalling the definition of derivative: a function is holomorphic at $z_0$ if $\lim\frac{f(z)-f(z_0)}{z-z_0}$ exists as $z\to z_0$. If you write out what this means in terms of $f$ as a function on $\mathbb{R}^2$, you'll see both that $f$ is differentiable in the real sense and also satisfies the Cauchy-Riemann equations, which gives a third way to think about a holomorphic or analytic function.

A fourth way that may be more accessible to intuition is the amplitwist concept from Visual Complex Analysis by Tristan Needham. The fundamental insight is simple and compelling: the derivative of a complex function at a point is just a complex number, so that holomorphic functions must act infinitesimally by rescaling and rotating, because that's all multiplication by a complex number does. In particular, holomorphic functions are conformal: they preserve angles between curves (this is a fifth, partially independent, way to think about complex differentiable functions.) I highly recommend Needham's book as you look for more insights into this subject.

Solution 2:

Holomorphic is just talking differentiability in the complex plane.

For example, in Real analysis, we look at the left and right limits only.

whereas here we are looking in all possible directions. Here we are giving eight of those possibilities in the diagram.

Analytic

Suppose a complex-valued function is holomorphic in all points of a domain then it is called as an analytic function. Hope this helps.

I hope the following link holomorphic will help for details.