- #1

- 533

- 1

Also, does it make sense to talk about functions being analytic/holomorphic at a POINT, or do we always need to talk about them being analytic in some NEIGHBORHOOD of a point (i.e., on an open set)?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter AxiomOfChoice
- Start date

- #1

- 533

- 1

Also, does it make sense to talk about functions being analytic/holomorphic at a POINT, or do we always need to talk about them being analytic in some NEIGHBORHOOD of a point (i.e., on an open set)?

- #2

- 811

- 6

The term analytic function is often used interchangeably with “holomorphic function”. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.

Being holomorphic at a point is also the same as saying it's holomorphic in some neighborhood of that point. (Just like being differentiable or continuous at a point).

- #3

Office_Shredder

Staff Emeritus

Science Advisor

Gold Member

- 4,527

- 572

http://en.wikipedia.org/wiki/Holomorphic_function

Being holomorphic at a point is also the same as saying it's holomorphic in some neighborhood of that point. (Just like being differentiable or continuous at a point).

A function can actually be continuous or differentiable at a point without being continuous or differentiable in a neighborhood. One example is f(x)=x

- #4

Petek

Gold Member

- 364

- 9

Also, does it make sense to talk about functions being analytic/holomorphic at a POINT, or do we always need to talk about them being analytic in some NEIGHBORHOOD of a point (i.e., on an open set)?

The function [itex]f(z) = |z|^2[/itex] has a derivative at z = 0, but not at any other point (verification is left as an exercise). Therefore, although the derivative exists at z = 0, the function isn't analytic at z = 0.

Petek

Share: