How many definitions of holomorphic

[cheeez]

There are a lot of definitions but what is the quickest way to see if a function is holomorphic? apply the cauchy riemann equations seems too slow. I thought if it doesn't have a z_bar in it, then it's automatically holomorphic. so for ex. polynomials are always holomorphic. on the other hand, 1/z on the unit circle is z_bar so it's not holomorphic but if you take the z_bar derivative of 1/z in the elementary calculus sense, it is 0. So as long as z can't be rewritten as z_bar it's fine to treat it as a constant? Are there things of this sort to watch out for?

 

[micromass]

There are a lot of disguises for a particular function, so seeing whether a function is holomorphic depends on what form the function takes. Somethimes it's quicker to apply Cauchy-Riemann, in other instances you need to do something else.

And it is not true that you only need to check whether z_bar enters into the equations. This may be a good intuitive way to check whether the function is holomorphic, but you'll still need to check it rigourously. In fact, there are a lot of functions which don't have z_bar in them and which are not holomorphic...

 

[Landau]

And it is not true that you only need to check whether z_bar enters into the equations. This may be a good intuitive way to check whether the function is holomorphic, but you'll still need to check it rigourously. In fact, there are a lot of functions which don't have z_bar in them and which are not holomorphic...

Well, it's true if you mean the right thing: the Cauchy-Riemann equations are equivalent to

$$\frac{\partial f}{\partial\bar{z}} = 0$$

where

$$\frac{\partial}{\partial\bar{z}}= \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)$$

 

[cheeez]

exactly what are those other functions without z_bar but are still not holomorphic because unless you apply the operator defined by landau it's very hard to tell. Seems like you have to use that operator to make sure. is there anyway to tell on inspection if a function is holomorphic, maybe some geometric intuition?

also why is it that z^1/3 is not a well defined holomorphic function in any neighborhood of 0.