Re: entire~~

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 06/07/04 

Date: Mon, 07 Jun 2004 18:56:04 +0100

mina_world wrote:

> hello.....sir~
>
>
> Prove that the function f defined by means of the
>
> equations
>
> f(z) =
>
> (e^z) - 1 when z =/= 0
>
> 1 when z = 0
>
> is entire.
>

Let f be analytic on a domain U. Let a in U and define
g(z) = (f(z) - f(a))/(z - a) for z in U but z =/= a.
     = f'(a) if z = a.

Then g is analytic on U. This comes from the basic theory
of removable singularities. 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_