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I've been asked to find the number of distinct values that $ \oint_\gamma \frac{dz}{(z-a_1)(z-a_2)...(z-a_n)}$ can take for simple closed curves $ \gamma $ not passing through any of the $ a_i $.

My thoughts so far:

I know I should be thinking about partial fractions and using Cauchy's Integral Formula. WLOG set $ |a_1| < |a_2| < ... < |a_n| $. Let $ I(\gamma) $ denote the interior of the closed curve $ \gamma $. Now there are $ n $ possible scenarios that could yield different values for the integral: $ a_1 \in I(\gamma) $, $ a_1 $ and $ a_2 \in I(\gamma) $, ... , $ a_1, a_2, ... , a_n \in I(\gamma) $. From the integral formula, each root $ a_i $ of $ p(z) = (z-a_1)...(z-a_n) $ will make a non-zero contribution iff $ a_i \in I(\gamma) $. Writing as partial fractions, we have:

$ \frac{1}{p(z)} = \frac{A_1}{z-a_1} + \frac{A_2}{z - a_2} + ... + \frac{A_n}{z - a_n} $, which leads to

$ A_1 (z-a_2)...(z-a_n) + A_2(z-a_1)...(z -a_n) + ... + A_n(z-a_1)...(z-a_{n-1}) = 1$ (*)

And so we see that the possible values of the integral are $ 2\pi i \left( A_1 \right) $, or $ 2\pi i \left( A_1 + A_2 \right) $, or ... $ 2 \pi i \left( A_1 + ... + A_n \right) $.

But the relationship (*) gives restrictions on the values of the $ A_i $. For example, $ A_1 + ... + A_n = 0 $ (by considering the coefficient of $ z^{n+1} $).

I'm unsure about making the final leap to the answer - how should I count the possible values of the $n $ sums? I would greatly appreciate any advice, either in the form of pointing out mistakes in my reasoning so far or hints on where to go next. I'd rather not have a full answer (or a hint that makes my task trivial). Thanks.

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I've had another thought: as soon as you fix $ A_n $, the other possible values of the integral are all fixed also. So the question (I think) widdles down to "how many possible values of $ A_n $ are there?". I think. –  Mathmo Apr 30 '11 at 19:19
    
It is true that when $\gamma$ goes around all the pole that the answer is zero. You can see that this is the case because you can imagine the complex plane to be a sphere (Riemann sphere) and then you can take the complement of $I(\gamma)$ (which has no poles inside) as the interior... –  Fabian Apr 30 '11 at 19:20
    
@Fabian: I don't understand what you're saying. What about $n=1$ and $a_1 = 0$? –  t.b. Apr 30 '11 at 19:22
    
I think the residue theorem is useful here. –  Jose27 Apr 30 '11 at 19:25
    
I guess I missed something, but why don't you consider the case when the interior of the loop $I(\gamma)$ contains $a_2$ and not $a_1$, just to name one? –  Raskolnikov Apr 30 '11 at 19:27

2 Answers 2

That $\sum_i A_i=0$ is the only condition the $A_i$'s generically have to fulfill. Thus, the value of the integral can be $$A_1, A_2, \dots, A_n,$$ $$A_1 + A_2, A_1 + A_3, \dots, A_{n-1}+A_n,$$ $$\dots$$ $$A_{1}+A_2 + \dots + A_{n-1} =-A_{n}, -A_{n-1}, \dots , -A_1.$$

Of the first line there are $n={n\choose 1}$ distinct values. The second line yields ${n\choose 2}$ distinct values. In total, we have $$\sum_{k=0}^{n-1} {n \choose k} = 2^n -1$$ distinct values of the integral; $k=0$ counts the value zero which you get when you enclose nothing or everything. Note that for $n=1$ the formula does not work. As there are obviously two results (0 or $A_1$) possible.

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Isn't this off by $1$? If $n=2$, the contour can enclose both, either, or neither, which is $4$ cases; both and neither give zero, so that's $3$ different values; your formula gives $2$. –  Gerry Myerson May 1 '11 at 4:30
    
@Gerry Myerson: you are right. I forgot that zero is also a number. I corrected the text. Thank you very much! –  user10287 May 1 '11 at 8:15

(this is a little correction of user10287's answer - we need to care about the orientation of the curve! so we get twice as many possibilities. It's the residue at infinity being $0$ that cuts the number back by $2$)

The curve divides the Riemann sphere to two parts. The integral is given by the sum of residues which are on the positive side of the curve. The residue at $\infty$ is $0$ (unless $n=1$!). So we get $2^n-1$ possibilities (or $2$ in the case $n=1$) ($2^n$ for the choice of the subset of $a_1,\dots,a_n$ which is on the positive side of the curve, and $-1$ for the case when nothing is on the positive side, giving the same answer as if everything is at on the positive side. It wouldn't fit as a comment, so I post it as an answer).

It is now a question whether all the remaining $2^n-1$ possibilities are different. The value of $A_1$ is $1/((a_2-a_1)(a_3-a_1)\dots(a_n-a_1))$ (multiply your equation by $z-a_1$ and then substitute $z=a_1$) (and similarly for other $A_i$'s). The question is whether there is any non-trivial identity $A_{i_1}+\dots+A_{i_k}=A_{j_1}+\dots+A_{j_l}$ besides $A_1+\dots +A_n=0$. It certainly depends on $a_i$'s - and for the moment I don't see why there is no such identity :)

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The question asks for the number of values the integral can take - I interpret that to mean, the maximum. I'm confident you can choose the $a_i$ so there are no other relations. –  Gerry Myerson May 1 '11 at 12:47

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