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?


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.


Holomorphic is just talking differentiability in complex plane....

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

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

Not holomorphic


Suppose a complex valued function is holomorphic in all points of a domain then it is called as Analytic function... Hope will help..

I hope the following link holomorphic will help for details..

  • $\begingroup$ If I may make a slightly off-topic comment, the profusion of ellipses is a bit distracting-and doesn't seem to communicate anything very specific. $\endgroup$ – Kevin Carlson Nov 21 '14 at 17:09
  • $\begingroup$ i could not understand what you are concluding from this... please be frank and give details.. hope will help me to learn correctly... please..@Kevin $\endgroup$ – David Nov 21 '14 at 17:18
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    $\begingroup$ I was just suggesting you use "..." less and stick to the punctuation "." for most sentences. It wasn't a mathematical comment. To make a mathematical comment, your edit isn't very clear. It seems to imply the issue with holomorphicity is of directional derivatives in oblique directions, but differentiable functions on $\mathbb{R}^2$ already have this property. It's the Cauchy-Riemann equations that make holomorphic functions something special, yes? $\endgroup$ – Kevin Carlson Nov 21 '14 at 17:20
  • $\begingroup$ yes..you are right...@Kevin $\endgroup$ – David Nov 21 '14 at 17:24
  • $\begingroup$ Don't understand why this is disliked. Clear cut and straight to the point. $\endgroup$ – jamie Feb 26 at 18:54

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