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Sum of residues

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Sum of residues

Hello everyone !

Consider an analytic function, to be specific, I am mostly interested in things like: F(z)=e^zt*H(z)/P(z) where H and P are polonymials in z.

Now the polynomial P depends on other other parameters that can be varied. Let a and b be two zeros of P with multiplicity g_a and g_b
and such that, by varying certain parameters of P, one has a->b i.e. the two can become exactly degenerate for certain parameter values.

Now let Res_a(F(z)) be the residue of F in a and Res_b(F(z)) the residue of F in b and let Res_{a,b}(F) be the residue of F in b when a and b are exactly degenerate.

What is the best bound for the following :
|Res_a(F)+Res_b(F)- Res_{a,b}(F)|

Indeed, playing around with polynoms you will notice that when two poles of a function approach each other, the residues at these poles become very large but, when added, then mostly cancel out and leave a small quantity which goes smoothly to Res_{a,b}(F) as expected.

I suspect this must be a very general observation, since one expect a continuous, smooth transition of the sum of residues from the situation where poles are nearly degenerate to that where they are exactly degenerate.

So what is the function D of the distance between the poles (a-b) such that

|Res_a(F)+Res_b(F)- Res_{a,b}(F)|=< D(a-b) ???

Clearly D(0)=0, but that's all I know. Any ideas ?


Alright it's obvious in fact. Express F as a function of d the distance between the poles. Then Taylor expansion that. If F(z)=f(z)/(z-(a+d))^g_b then you get

Res_a(F)+Res_b(F)=sum_n (g_b)_n d^n/n! Res_a(f(z),g_a+g_b+n)

with (g_b)_n the Pochhammer symbol and Res_a(f(z),g_a+g_b+n) the residue of f(z) in a with degeneracy g_a+g_b+n.

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