Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quick/easy question about analytic (holomorphic) functions

  1. Oct 4, 2009 #1
    Is saying "[itex]f[/itex] is differentiable" equivalent to saying "[itex]f[/itex] is analytic/holomorphic?"

    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. jcsd
  3. Oct 5, 2009 #2

    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).
  4. Oct 5, 2009 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    A function can actually be continuous or differentiable at a point without being continuous or differentiable in a neighborhood. One example is f(x)=x2 if x is rational, 0 if x is irrational.
  5. Oct 5, 2009 #4
    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.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook