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From: johnson542 on 22 Nov 2009 23:27
Is the following statement valid?

Let f(z) be an analytic function with an isolated essential
singularity at a, say f is analytic in the open disk D
0<|z-a|<r.
Then for any circle C with radius less than r,

int_C f(z) dz =2pi i b,

where b is the coefficient of 1/(z-a)
in the Laurant expansion of f in D. I.e. the Residue theorem
is valid at a.

Most books have this theorem for poles. That is why I am not sure if the above statement is true. If not, what is a counterexample?
From: Robert Israel on 23 Nov 2009 12:42

> Is the following statement valid?
>
> Let f(z) be an analytic function with an isolated essential
> singularity at a, say f is analytic in the open disk D
> 0<|z-a|<r.

punctured disk, not disk

> Then for any circle C with radius less than r,
>
> int_C f(z) dz =2pi i b,
>
> where b is the coefficient of 1/(z-a)
> in the Laurant expansion of f in D. I.e. the Residue theorem
> is valid at a.
>
> Most books have this theorem for poles. That is why I am not sure if the
> above statement is true. If not, what is a counterexample?

Yes, it is true.
More generally, the coefficient of (z-a)^k in the Laurent series for
f(z) in an annulus centred at a and containing the circle C around a
is 1/(2 pi i) int_C f(z) (z-a)^(-1-k) dz.
This should be a prominent piece of the exposition of Laurent series in any
complex variables text. No need to restrict it to poles.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: David C. Ullrich on 24 Nov 2009 07:07
On Mon, 23 Nov 2009 09:27:23 EST, johnson542 <johnson542(a)verizon.net>
wrote:

>Is the following statement valid?
>
>Let f(z) be an analytic function with an isolated essential
>singularity at a, say f is analytic in the open disk D
>0<|z-a|<r.
>Then for any circle C with radius less than r,
>
>int_C f(z) dz =2pi i b,
>
>where b is the coefficient of 1/(z-a)
>in the Laurant expansion of f in D. I.e. the Residue theorem
>is valid at a.

Certainly that's true.

>Most books have this theorem for poles.

One of many reasons you need a copy of "Complex Made Simple".

My guess is that the reason many books restrict the Residue Theorem to
functions with poles is that that's often sufficient, and also there
are little tricks to calculate the residue at a pole of order <= n
that don't work for an essential singularity.

>That is why I am not sure if the above statement is true. If not, what is a counterexample?

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
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