I've just been reading about how complex functions can be defined on the extended complex plane. They start with rational functions as examples, and defining them at oo so they're continuous at oo in a sense. Eg, 1/z would be defined to be 0 at z = oo.(adsbygoogle = window.adsbygoogle || []).push({});

I understand that given a holomorphic function f, then f is not a local homeomorphism whenever f'(z) = 0 (open mapping theorem right?), but I'm wondering, what if f'(z) = oo now? Or what if I get a situation where f'(z) isn't even defined at oo, which seems to happen a lot? For example, f(z) = z^2 isn't differentiable at oo if I understand this stuff correctly, but is differentiable everywhere else. I get the feeling that, since oo is a fixed point of z^2, and z^2 is going to be covering everything twice near oo, it's not going to be a local homeomorphism there either .. but I'm wondering if anyone could clear this up for me properly, or point me in the direction of some notes/books that deal with this stuff nicely.

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Local homeomorphism question

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Local homeomorphism question |
---|

I Divergent series question |

B Function rules question |

B Question about a limit definition |

A Angular Moment Operator Vector Identity Question |

I A question regarding Logistic population model |

**Physics Forums | Science Articles, Homework Help, Discussion**