calculuscomplex numberscomplex-analysisdefinition
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?
The two notions of being differentiable used here, $\mathbb R$-differentiability and $\mathbb C$-differentiability, are not the same. The definitions may look similar, but the implications of these definitions are very different.
A diffeomorphism is required to have a nonvanishing derivative. The derivative of a holomorphic function is allowed to be zero.
A diffeomorphism is required to be a bijection. A holomorphic function does not have to be.
Consider the function $f(x) = \mathrm{e}^{-1/x^2}$ on the reals. Although not defined at zero, it clearly has a limit at zero and that limit is zero. Now look at its derivatives: $$f'(x) = \frac{2}{x^3} f(x)$$ $$f''(x) = \frac{4-6x^2}{x^6} f(x)$$ $$f'''(x) = \frac{8-36x^2+24x^4}{x^9} f(x)$$ and so on, with the exponential overwhelming the rational function as we take limits at zero. We find, in fact, that this limiting process gives us zeroes for every derivative at zero.
Is this the zero function? No. Therefore, its power series expansion is not faithful at zero.
Why? Take a step back and look at the complex plane and notice the essential singularities pressed up against the real line at zero and discover that I "cheated" by taking limits at zero. In the reals, the limits exist, but in the complexes they do not. In short, the real line can be blind to essential features of functions. (Especially if the person posing functions to investigate is aware of this deficiency.) However, having looked at a neighborhood in the complexes, there is nothing left to know since there is nowhere for a feature to hide.
(I spent some time trying to clean up what amounts to random moire pattern in the center of the argument plot. The best I could manage follows, but is misleading about the violent flailing of the complex angle near the origin.)
(What we should really be seeing is infinitely many thinner and thinner strips as we get closer to the origin. Of course, the strips are already one pixel wide in the given image -- if we attempt to thin them further, all we render is a continuous blob of contour lines.)
Best Answer
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.