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do you know if it's possible to find the number of zeros on an open set of quaternions via one integral?

There's a way in the set of complex numbers. For f being holomorphic in G and behind its boundary we get

$$ N_0 (f, G)=\int_{\partial G}\frac{f'(z)}{f(z)}dz. $$ Does it exist an analogical one for $\mathbb{H}$? Or maybe is there another way? The considered functions look like Taylor series with coefitiens always on the left. Thank you all, guys, for any help

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    $\begingroup$ Hi, welcome to Math SE. Bear in mind $z^2+1$ has $\beth$ zeroes, which fit in an open set. You can't express that cardinality as an integral. $\endgroup$ Commented Nov 2 at 15:54
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    $\begingroup$ You're right. We get every strictly imaginary number of norm 1 if we take G to be ball of radius 2. But nowhere have i found an answer if for some cases it's been impossible, searching books. Remember, analogically Hopf-Poincare theorem does not "see" zeros if they're not isloated. That is why i still have a little hope heh $\endgroup$ Commented Nov 2 at 16:30

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