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7.3 Power series from complex smoothness
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Joined: 14 Sep 2011, 03:38
Posts: 2
This section (7.3) describes the Cauchy formula and begins "The above displayed expression is a particular case ( for the constant function f(z)=2*pi*i ) of the famous Cauchy formula..."

I take the "above displayed expression" to mean the result for the contour integral of dz/z which is given as 2*pi*i on the preceding page (next to last paragraph of 7.2).

But the Cauchy formula as presented immediately following the first paragraph in 7.3 seems to me to require that the "particular case" be for the constant function f(z) = 1, not the constant function f(z)=2*pi*i.

Can someone clarify this for me?

14 Sep 2011, 04:21 Supporter Joined: 07 Jun 2008, 08:21
Posts: 235
You are almost right and Penrose is right!

Penrose takes f(z)=2*pi*i where f(0)=2*pi*i
The Cauchy formula in this case is the integral round the origin of dz/z, which evaluates to 2*pi*i=f(0) as on page 126

you take f(z)=1 where f(0)=1
The Cauchy formula in this case evaluates to 1=f(0)

So you are both right - kind of!

The point is that for any constant function of z, say f(z)=k (a complex constant) the Cauchy formula evaluates to k=f(0).

Dividing by 1/(2*pi*i) means that the Cauchy formula always evaluates to f(0), otherwise it would evaluate to 2*pi*i*f(0).
I hope this helps.
Vasco

14 Sep 2011, 07:18 Joined: 14 Sep 2011, 03:38
Posts: 2
Vasco,

Thank you for this clarification. It helps me a lot with the whole chapter.

I'm going to support this forum, as I am having the time of my life with this wonderful book, and will surely need more help along the way.

Frank

14 Sep 2011, 17:04 Page 1 of 1 [ 3 posts ] 