How many definitions of holomorphic

  • Context: Graduate 
  • Thread starter Thread starter cheeez
  • Start date Start date
  • Tags Tags
    Definitions
Click For Summary

Discussion Overview

The discussion revolves around the definitions and criteria for determining whether a function is holomorphic. Participants explore various methods for identifying holomorphic functions, including the Cauchy-Riemann equations and intuitive checks related to the presence of the conjugate variable z_bar. The conversation includes theoretical considerations and practical implications of these definitions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that if a function does not contain z_bar, it may be automatically considered holomorphic, using polynomials as an example.
  • Another participant counters that the absence of z_bar does not guarantee holomorphicity, emphasizing the need for rigorous checks, including the Cauchy-Riemann equations.
  • A later reply reiterates the importance of the Cauchy-Riemann equations, stating they are equivalent to the condition that the derivative with respect to z_bar equals zero.
  • One participant questions what specific functions lack z_bar but are still not holomorphic, indicating the complexity of identifying such functions without rigorous methods.
  • There is a query about the geometric intuition for determining holomorphicity and a specific mention of the function z^(1/3) not being well-defined as holomorphic near zero.

Areas of Agreement / Disagreement

Participants express disagreement regarding the sufficiency of checking for z_bar in determining holomorphicity. There is no consensus on a singular method for identifying holomorphic functions, with multiple perspectives on the criteria and approaches involved.

Contextual Notes

Participants note that certain functions may be challenging to classify as holomorphic without applying specific operators or rigorous definitions, highlighting the limitations of intuitive checks.

Who May Find This Useful

This discussion may be of interest to students and professionals in mathematics and physics, particularly those studying complex analysis and the properties of holomorphic functions.

cheeez
Messages
20
Reaction score
0
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?
 
Physics news on Phys.org
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...
 
micromass said:
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

[tex]\frac{\partial f}{\partial\bar{z}} = 0[/tex]

where

[tex]\frac{\partial}{\partial\bar{z}}= \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)[/tex]
 
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.
 

Similar threads

How Do I Solve These Complex Contour Integrals?
  • · Replies 1 ·
Replies
1
Views
2K
Holomorphic function reduces to a polynomial
  • · Replies 3 ·
Replies
3
Views
2K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
Graduate What is the definition of integration over a manifold and what does it measure?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad A couple questions about the Riemann Tensor, definition and convention
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Representing a factorial through its pseudo Z transform
  • · Replies 1 ·
Replies
1
Views
2K
Understanding the Cauchy Integration Formula for Analytic Functions
  • · Replies 4 ·
Replies
4
Views
2K
Graduate How Can the Residue Theorem Be Applied to Prove This Integral?
  • · Replies 1 ·
Replies
1
Views
3K
Challenge Math Challenge - October 2020
  • · Replies 33 ·
2
Replies
33
Views
9K
How Do You Evaluate sin(x)/x^4 Over the Unit Circle Using Complex Analysis?
  • · Replies 3 ·
Replies
3
Views
2K